{"id":6896,"date":"2026-09-14T14:01:33","date_gmt":"2026-09-14T08:31:33","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=6896"},"modified":"2026-09-14T14:20:30","modified_gmt":"2026-09-14T08:50:30","slug":"class-12-maths-exercise-5-2-continuity-and-differentiability","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/class-12-maths-exercise-5-2-continuity-and-differentiability\/","title":{"rendered":"Class 12 | Maths | Exercise 5.2 \u2013 Continuity and Differentiability"},"content":{"rendered":"<div class=\"mymoc-top mymoc-entity-placement\" id=\"mymoc-1157607251\"><div id=\"mymoc-2911340015\"><a href=\"https:\/\/amzn.to\/440rYSB\" aria-label=\"ssd\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/ssd.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/ssd.png 1303w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/ssd-300x50.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/ssd-1024x171.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/ssd-768x128.png 768w\" sizes=\"(max-width: 1303px) 100vw, 1303px\" width=\"1303\" height=\"218\"><\/a><\/div><\/div>\n<h3 class=\"wp-block-heading\">Differentiation Using the Chain Rule | NCERT Class 12 Mathematics<\/h3>\n\n\n\n<h2 class=\"wp-block-heading\">Quick Facts<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Exercise 5.2<\/strong> focuses mainly on differentiation of composite functions and differentiability.<\/li>\n\n\n\n<li>The <strong>Chain Rule<\/strong> is used when one function is inside another function.<\/li>\n\n\n\n<li>If <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>u<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y=f(u)<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">u=g(x)<\/annotation><\/semantics><\/math>, then:<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><\/mfrac><mo>&sdot;<\/mo><mfrac><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dy}{dx}=\\frac{dy}{du}\\cdot\\frac{du}{dx}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Standard derivatives frequently used in this exercise:<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dx}(\\sin x)=\\cos x<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>&minus;<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dx}(\\cos x)=-\\sin x<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>tan<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mrow><mi>sec<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dx}(\\tan x)=\\sec^2x<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>cot<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>&minus;<\/mo><msup><mrow><mi>csc<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dx}(\\cot x)=-\\csc^2x<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>sec<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>sec<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><mi>tan<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dx}(\\sec x)=\\sec x\\tan x<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>n<\/mi><msup><mi>x<\/mi><mrow><mi>n<\/mi><mo>&minus;<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dx}(x^n)=nx^{n-1}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A function can be <strong>continuous but not differentiable<\/strong> at a point, as illustrated by modulus and greatest integer functions in this exercise.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Story<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Imagine a function as a machine. Sometimes <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math> goes directly into the machine:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y=\\sin x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But sometimes <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math> first enters one machine and its output enters another:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>x<\/mi><mo>&rarr;<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mo>&rarr;<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">x\\rightarrow x^2+5\\rightarrow \\sin(x^2+5)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here, the function is <strong>composite<\/strong>. To differentiate it, we need to follow the same journey in reverse.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For example,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y=\\sin(x^2+5)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/relations-and-functions-exercise-1-1-solutions-class-12\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">Relations and Functions Exercise 1.1 Solutions Class 12<\/span><\/a><\/div><p class=\"wp-block-paragraph\">First differentiate the outside function:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cos(x^2+5)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then multiply by the derivative of the inside function:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{dy}{dx}=2x\\cos(x^2+5)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This step-by-step process is the <strong>Chain Rule<\/strong>, which the textbook introduces for composite functions.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Key Terms<\/h1>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Term<\/th><th>Meaning<\/th><\/tr><\/thead><tbody><tr><td><strong>Composite Function<\/strong><\/td><td>A function formed by applying one function to another<\/td><\/tr><tr><td><strong>Chain Rule<\/strong><\/td><td>Rule used to differentiate composite functions<\/td><\/tr><tr><td><strong>Inner Function<\/strong><\/td><td>The function inside another function<\/td><\/tr><tr><td><strong>Outer Function<\/strong><\/td><td>The function acting on the inner function<\/td><\/tr><tr><td><strong>Differentiability<\/strong><\/td><td>Existence of a finite derivative at a point<\/td><\/tr><tr><td><strong>Left-hand Derivative<\/strong><\/td><td>Derivative obtained by approaching a point from the left<\/td><\/tr><tr><td><strong>Right-hand Derivative<\/strong><\/td><td>Derivative obtained by approaching a point from the right<\/td><\/tr><tr><td><strong>Greatest Integer Function<\/strong><\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">[<\/mo><mi>x<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">[x]<\/annotation><\/semantics><\/math>, the greatest integer less than or equal to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td><strong>Modulus Function<\/strong><\/td><td>(<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The textbook states that a function is differentiable at a point when the relevant left- and right-hand derivative limits are finite and equal.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Questions &ndash; Detailed Solutions<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">The PDF contains <strong>10 questions in Exercise 5.2<\/strong>: Questions 1&ndash;8 ask for differentiation, while Questions 9&ndash;10 ask for proofs of non-differentiability.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 1<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Differentiate:<\/h3>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/relations-and-functions-exercise-1-2-solutions-class-12\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">Relations and Functions Exercise 1.2 Solutions Class 12<\/span><\/a><\/div><p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y=\\sin(x^2+5)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">u=x^2+5<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mi>u<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y=\\sin u<\/annotation><\/semantics><\/math><\/p><div class=\"mymoc-middle mymoc-entity-placement\" id=\"mymoc-1600226806\"><div id=\"mymoc-635440936\"><a href=\"https:\/\/amzn.to\/4upqwUD\" aria-label=\"laptop-1\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/laptop-1.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/laptop-1.png 1303w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/laptop-1-300x91.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/laptop-1-1024x310.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/laptop-1-768x233.png 768w\" sizes=\"(max-width: 1303px) 100vw, 1303px\" width=\"1303\" height=\"395\"><\/a><\/div><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Using the Chain Rule,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mi>u<\/mi><mfrac><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dy}{dx}=\\cos u\\frac{du}{dx}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{du}{dx}=2x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{dy}{dx}=2x\\cos(x^2+5)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 2<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Differentiate:<\/h3>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/miscellaneous-exercise-solutions-class-12-relations-functions\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">Miscellaneous Exercise Solutions Class 12 Relations Functions<\/span><\/a><\/div><p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y=\\cos(\\sin x)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">u=\\sin x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mi>u<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y=\\cos u<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>&minus;<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mi>u<\/mi><mfrac><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dy}{dx}=-\\sin u\\frac{du}{dx}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{du}{dx}=\\cos x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">we get<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>&minus;<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{dy}{dx}=-\\cos x\\sin(\\sin x)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 3<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Differentiate:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y=\\sin(ax+b)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">u=ax+b<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mi>u<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y=\\sin u<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using the Chain Rule,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mi>u<\/mi><mfrac><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dy}{dx}=\\cos u\\frac{du}{dx}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{du}{dx}=a<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">we obtain<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>a<\/mi><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{dy}{dx}=a\\cos(ax+b)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 4<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Differentiate:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>sec<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>tan<\/mi><mo>&#8289;<\/mo><msqrt><mi>x<\/mi><\/msqrt><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y=\\sec(\\tan\\sqrt{x})<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This contains <strong>three layers<\/strong>:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>x<\/mi><mo>&rarr;<\/mo><msqrt><mi>x<\/mi><\/msqrt><mo>&rarr;<\/mo><mi>tan<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msqrt><mi>x<\/mi><\/msqrt><mo stretchy=\"false\">)<\/mo><mo>&rarr;<\/mo><mi>sec<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>tan<\/mi><mo>&#8289;<\/mo><msqrt><mi>x<\/mi><\/msqrt><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">x\\rightarrow\\sqrt{x}\\rightarrow\\tan(\\sqrt{x})\\rightarrow\\sec(\\tan\\sqrt{x})<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><msqrt><mi>x<\/mi><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">u=\\sqrt{x}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>v<\/mi><mo>=<\/mo><mi>tan<\/mi><mo>&#8289;<\/mo><mi>u<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">v=\\tan u<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>sec<\/mi><mo>&#8289;<\/mo><mi>v<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y=\\sec v<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>v<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>sec<\/mi><mo>&#8289;<\/mo><mi>v<\/mi><mi>tan<\/mi><mo>&#8289;<\/mo><mi>v<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dy}{dv}=\\sec v\\tan v<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mrow><mi>sec<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>u<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dv}{du}=\\sec^2u<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><msqrt><mi>x<\/mi><\/msqrt><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{du}{dx}=\\frac{1}{2\\sqrt{x}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>sec<\/mi><mo>&#8289;<\/mo><mi>v<\/mi><mi>tan<\/mi><mo>&#8289;<\/mo><mi>v<\/mi><mo>&sdot;<\/mo><msup><mrow><mi>sec<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>u<\/mi><mo>&sdot;<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><msqrt><mi>x<\/mi><\/msqrt><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dy}{dx} = \\sec v\\tan v\\cdot\\sec^2u\\cdot\\frac{1}{2\\sqrt{x}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Substituting back,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>sec<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>tan<\/mi><mo>&#8289;<\/mo><msqrt><mi>x<\/mi><\/msqrt><mo stretchy=\"false\">)<\/mo><mi>tan<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>tan<\/mi><mo>&#8289;<\/mo><msqrt><mi>x<\/mi><\/msqrt><mo stretchy=\"false\">)<\/mo><msup><mrow><mi>sec<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><msqrt><mi>x<\/mi><\/msqrt><\/mrow><mrow><mn>2<\/mn><msqrt><mi>x<\/mi><\/msqrt><\/mrow><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{ \\frac{dy}{dx} = \\frac{\\sec(\\tan\\sqrt{x})\\tan(\\tan\\sqrt{x})\\sec^2\\sqrt{x}} {2\\sqrt{x}} }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 5<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Differentiate:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mfrac><mrow><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>x<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">y=\\frac{\\sin(ax+b)}{\\cos(cx+d)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Use the <strong>Quotient Rule<\/strong>:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>u<\/mi><mi>v<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mfrac><mrow><mi>v<\/mi><msup><mi>u<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msup><mo>&minus;<\/mo><mi>u<\/mi><msup><mi>v<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msup><\/mrow><msup><mi>v<\/mi><mn>2<\/mn><\/msup><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dx}\\left(\\frac{u}{v}\\right) = \\frac{vu&rsquo;-uv&rsquo;}{v^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Take<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">u=\\sin(ax+b)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>v<\/mi><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>x<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">v=\\cos(cx+d)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>u<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msup><mo>=<\/mo><mi>a<\/mi><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">u&rsquo;=a\\cos(ax+b)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>v<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msup><mo>=<\/mo><mo>&minus;<\/mo><mi>c<\/mi><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>x<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">v&rsquo;=-c\\sin(cx+d)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>x<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mtext>&thinsp;<\/mtext><mi>a<\/mi><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>&minus;<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mrow><mo fence=\"true\">[<\/mo><mo>&minus;<\/mo><mi>c<\/mi><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>x<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><mrow><msup><mrow><mi>cos<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>x<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dy}{dx} = \\frac{ \\cos(cx+d)\\,a\\cos(ax+b) &ndash; \\sin(ax+b)\\left[-c\\sin(cx+d)\\right] } {\\cos^2(cx+d)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>a<\/mi><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>x<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>c<\/mi><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>x<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msup><mrow><mi>cos<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>x<\/mi><mo>+<\/mo><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{ \\frac{dy}{dx} = \\frac{ a\\cos(ax+b)\\cos(cx+d) +c\\sin(ax+b)\\sin(cx+d) } {\\cos^2(cx+d)} }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 6<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Differentiate:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo>&sdot;<\/mo><msup><mrow><mi>sin<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y=\\cos x^3\\cdot\\sin^2(x^5)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using the <strong>Product Rule<\/strong>:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>u<\/mi><mi>v<\/mi><msup><mo stretchy=\"false\">)<\/mo><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msup><mo>=<\/mo><msup><mi>u<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msup><mi>v<\/mi><mo>+<\/mo><mi>u<\/mi><msup><mi>v<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(uv)&rsquo;=u&rsquo;v+uv&rsquo;<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">u=\\cos(x^3)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>v<\/mi><mo>=<\/mo><msup><mrow><mi>sin<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">v=\\sin^2(x^5)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">First,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>u<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msup><mo>=<\/mo><mo>&minus;<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo>&sdot;<\/mo><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">u&rsquo;=-\\sin(x^3)\\cdot3x^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">so<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>u<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msup><mo>=<\/mo><mo>&minus;<\/mo><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">u&rsquo;=-3x^2\\sin(x^3)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>v<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">v<\/annotation><\/semantics><\/math>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>v<\/mi><mo>=<\/mo><mo stretchy=\"false\">[<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><msup><mo stretchy=\"false\">]<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">v=[\\sin(x^5)]^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>v<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msup><mo>=<\/mo><mn>2<\/mn><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo>&sdot;<\/mo><mn>5<\/mn><msup><mi>x<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">v&rsquo;=2\\sin(x^5)\\cos(x^5)\\cdot5x^4<\/annotation><\/semantics><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>v<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msup><mo>=<\/mo><mn>10<\/mn><msup><mi>x<\/mi><mn>4<\/mn><\/msup><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">v&rsquo;=10x^4\\sin(x^5)\\cos(x^5)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using the Product Rule,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>&minus;<\/mo><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><msup><mrow><mi>sin<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>10<\/mn><msup><mi>x<\/mi><mn>4<\/mn><\/msup><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dy}{dx} = -3x^2\\sin(x^3)\\sin^2(x^5) + 10x^4\\cos(x^3)\\sin(x^5)\\cos(x^5)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>&minus;<\/mo><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><msup><mrow><mi>sin<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>10<\/mn><msup><mi>x<\/mi><mn>4<\/mn><\/msup><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>5<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{ \\frac{dy}{dx} = -3x^2\\sin(x^3)\\sin^2(x^5) +10x^4\\cos(x^3)\\sin(x^5)\\cos(x^5) }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 7<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Differentiate:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><msqrt><mrow><mi>cot<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">y=2\\sqrt{\\cot(x^2)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Rewrite:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><mo stretchy=\"false\">[<\/mo><mi>cot<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><msup><mo stretchy=\"false\">]<\/mo><mrow><mn>1<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>2<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">y=2[\\cot(x^2)]^{1\/2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using the Chain Rule,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>2<\/mn><mo>&sdot;<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">[<\/mo><mi>cot<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><msup><mo stretchy=\"false\">]<\/mo><mrow><mo>&minus;<\/mo><mn>1<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>2<\/mn><\/mrow><\/msup><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">[<\/mo><mi>cot<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dy}{dx} = 2\\cdot\\frac12[\\cot(x^2)]^{-1\/2} \\frac{d}{dx}[\\cot(x^2)]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">[<\/mo><mi>cot<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mo>&minus;<\/mo><msup><mrow><mi>csc<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo>&sdot;<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dx}[\\cot(x^2)] = -\\csc^2(x^2)\\cdot2x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>&minus;<\/mo><mfrac><mrow><mn>2<\/mn><mi>x<\/mi><msup><mrow><mi>csc<\/mi><mo>&#8289;<\/mo><\/mrow><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><msqrt><mrow><mi>cot<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><\/msqrt><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{ \\frac{dy}{dx} = -\\frac{2x\\csc^2(x^2)} {\\sqrt{\\cot(x^2)}} }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Question 8<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Differentiate:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msqrt><mi>x<\/mi><\/msqrt><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y=\\cos(\\sqrt{x})<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><msqrt><mi>x<\/mi><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">u=\\sqrt{x}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mi>u<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y=\\cos u<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>&minus;<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mi>u<\/mi><mfrac><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dy}{dx} = -\\sin u\\frac{du}{dx}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>u<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><msqrt><mi>x<\/mi><\/msqrt><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{du}{dx}=\\frac{1}{2\\sqrt{x}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">we get<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>&minus;<\/mo><mfrac><mrow><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msqrt><mi>x<\/mi><\/msqrt><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mn>2<\/mn><msqrt><mi>x<\/mi><\/msqrt><\/mrow><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{ \\frac{dy}{dx} = -\\frac{\\sin(\\sqrt{x})}{2\\sqrt{x}} }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 9<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Prove that<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi mathvariant=\"normal\">&#8739;<\/mi><mi>x<\/mi><mo>&minus;<\/mo><mn>1<\/mn><mi mathvariant=\"normal\">&#8739;<\/mi><mo separator=\"true\">,<\/mo><mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>&isin;<\/mo><mi mathvariant=\"double-struck\">R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=|x-1|,\\qquad x\\in\\mathbb R<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is <strong>not differentiable at x=1x=1<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>&lt;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x&lt;1<\/annotation><\/semantics><\/math>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">&#8739;<\/mi><mi>x<\/mi><mo>&minus;<\/mo><mn>1<\/mn><mi mathvariant=\"normal\">&#8739;<\/mi><mo>=<\/mo><mn>1<\/mn><mo>&minus;<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">|x-1|=1-x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>&gt;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x&gt;1<\/annotation><\/semantics><\/math>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">&#8739;<\/mi><mi>x<\/mi><mo>&minus;<\/mo><mn>1<\/mn><mi mathvariant=\"normal\">&#8739;<\/mi><mo>=<\/mo><mi>x<\/mi><mo>&minus;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">|x-1|=x-1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">At <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x=1<\/annotation><\/semantics><\/math>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(1)=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Left-hand derivative<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>f<\/mi><mo>&minus;<\/mo><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msubsup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munder><mrow><mi>lim<\/mi><mo>&#8289;<\/mo><\/mrow><mrow><mi>h<\/mi><mo>&rarr;<\/mo><msup><mn>0<\/mn><mo>&minus;<\/mo><\/msup><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>h<\/mi><mo stretchy=\"false\">)<\/mo><mo>&minus;<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mi>h<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">f&rsquo;_-(1) = \\lim_{h\\to0^-} \\frac{f(1+h)-f(1)}{h}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>h<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">h&lt;0<\/annotation><\/semantics><\/math>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>h<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi mathvariant=\"normal\">&#8739;<\/mi><mi>h<\/mi><mi mathvariant=\"normal\">&#8739;<\/mi><mo>=<\/mo><mo>&minus;<\/mo><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(1+h)=|h|=-h<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>f<\/mi><mo>&minus;<\/mo><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msubsup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munder><mrow><mi>lim<\/mi><mo>&#8289;<\/mo><\/mrow><mrow><mi>h<\/mi><mo>&rarr;<\/mo><msup><mn>0<\/mn><mo>&minus;<\/mo><\/msup><\/mrow><\/munder><mfrac><mrow><mo>&minus;<\/mo><mi>h<\/mi><\/mrow><mi>h<\/mi><\/mfrac><mo>=<\/mo><mo>&minus;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f&rsquo;_-(1) = \\lim_{h\\to0^-}\\frac{-h}{h} =-1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Right-hand derivative<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>h<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">h&gt;0<\/annotation><\/semantics><\/math>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>h<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(1+h)=h<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>f<\/mi><mo>+<\/mo><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msubsup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munder><mrow><mi>lim<\/mi><mo>&#8289;<\/mo><\/mrow><mrow><mi>h<\/mi><mo>&rarr;<\/mo><msup><mn>0<\/mn><mo>+<\/mo><\/msup><\/mrow><\/munder><mfrac><mi>h<\/mi><mi>h<\/mi><\/mfrac><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f&rsquo;_+(1) = \\lim_{h\\to0^+}\\frac{h}{h} =1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>f<\/mi><mo>&minus;<\/mo><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msubsup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo mathvariant=\"normal\">&ne;<\/mo><msubsup><mi>f<\/mi><mo>+<\/mo><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">&prime;<\/mo><\/msubsup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f&rsquo;_-(1)\\ne f&rsquo;_+(1)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore the derivative does not exist at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x=1<\/annotation><\/semantics><\/math>.<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mtext>Hence,&nbsp;<\/mtext><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi mathvariant=\"normal\">&#8739;<\/mi><mi>x<\/mi><mo>&minus;<\/mo><mn>1<\/mn><mi mathvariant=\"normal\">&#8739;<\/mi><mtext>&nbsp;is&nbsp;not&nbsp;differentiable&nbsp;at&nbsp;<\/mtext><mi>x<\/mi><mo>=<\/mo><mn>1.<\/mn><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Hence, }f(x)=|x-1|\\text{ is not differentiable at }x=1.}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This follows the textbook&rsquo;s treatment of the modulus function: continuity does not necessarily imply differentiability.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 10<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Prove that the greatest integer function<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">[<\/mo><mi>x<\/mi><mo stretchy=\"false\">]<\/mo><mo separator=\"true\">,<\/mo><mspace width=\"2em\"><\/mspace><mn>0<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=[x],\\qquad 0&lt;x&lt;3<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is not differentiable at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x=1<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x=2<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The greatest integer function gives the greatest integer less than or equal to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Around <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x=1<\/annotation><\/semantics><\/math>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>&lt;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x&lt;1<\/annotation><\/semantics><\/math> and sufficiently close to 1,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">[<\/mo><mi>x<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">[x]=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>&gt;<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x&gt;1<\/annotation><\/semantics><\/math> and sufficiently close to 1,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">[<\/mo><mi>x<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">[x]=1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><munder><mrow><mi>lim<\/mi><mo>&#8289;<\/mo><\/mrow><mrow><mi>h<\/mi><mo>&rarr;<\/mo><msup><mn>0<\/mn><mo>&minus;<\/mo><\/msup><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>h<\/mi><mo stretchy=\"false\">)<\/mo><mo>&minus;<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mi>h<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\lim_{h\\to0^-}\\frac{f(1+h)-f(1)}{h}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><munder><mrow><mi>lim<\/mi><mo>&#8289;<\/mo><\/mrow><mrow><mi>h<\/mi><mo>&rarr;<\/mo><msup><mn>0<\/mn><mo>+<\/mo><\/msup><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>h<\/mi><mo stretchy=\"false\">)<\/mo><mo>&minus;<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mi>h<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\lim_{h\\to0^+}\\frac{f(1+h)-f(1)}{h}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">cannot give the same finite value. Hence <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math> is not differentiable at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x=1<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Similarly, around <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x=2<\/annotation><\/semantics><\/math>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x&lt;2<\/annotation><\/semantics><\/math>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">[<\/mo><mi>x<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">[x]=1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">while for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>&gt;<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x&gt;2<\/annotation><\/semantics><\/math>,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">[<\/mo><mi>x<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">[x]=2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus the left-hand and right-hand derivatives at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x=2<\/annotation><\/semantics><\/math> are not equal.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mtext>The&nbsp;greatest&nbsp;integer&nbsp;function&nbsp;is&nbsp;not&nbsp;differentiable&nbsp;at&nbsp;<\/mtext><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><mtext>&nbsp;and&nbsp;<\/mtext><mi>x<\/mi><mo>=<\/mo><mn>2.<\/mn><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{The greatest integer function is not differentiable at }x=1\\text{ and }x=2.}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The textbook also explains that the greatest integer function has jumps at integral points, which is why it is discontinuous there.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Memory Tricks<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Trick 1: &ldquo;Outside &times; Inside&rdquo;<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y=f(g(x))<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">remember:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"wp-block-paragraph\"><strong>Derivative of Outside &times; Derivative of Inside<\/strong><\/p>\n<\/blockquote>\n\n\n\n<p class=\"wp-block-paragraph\">Example:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo>&times;<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dx}\\sin(x^2) = \\cos(x^2)\\times2x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Trick 2: Three-layer functions<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>sec<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><mi>tan<\/mi><mo>&#8289;<\/mo><msqrt><mi>x<\/mi><\/msqrt><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y=\\sec(\\tan\\sqrt{x})<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">read from <strong>outside to inside<\/strong>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>SEC &rarr; TAN &rarr; &radic;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then differentiate from <strong>inside to outside<\/strong>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>&radic; &rarr; TAN &rarr; SEC<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Trick 3: Modulus = Corner<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A modulus graph often creates a <strong>sharp corner<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">At a corner:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mtext>LHD<\/mtext><mo mathvariant=\"normal\">&ne;<\/mo><mtext>RHD<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\text{LHD}\\ne\\text{RHD}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the derivative does not exist.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Trick 4: Greatest Integer = Jump<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Remember:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"wp-block-paragraph\"><strong>GI function jumps at integers.<\/strong><\/p>\n<\/blockquote>\n\n\n\n<p class=\"wp-block-paragraph\">At a jump, differentiability fails.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Case Study<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Case Study: Modelling the Position of a Moving Object<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Suppose the position of an object is represented by<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>sin<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">s(t)=\\sin(t^2+5)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t<\/annotation><\/semantics><\/math> represents time.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To determine its instantaneous velocity, differentiate <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">s(t)<\/annotation><\/semantics><\/math>:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>v<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi>d<\/mi><mi>s<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">v(t)=\\frac{ds}{dt}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using the Chain Rule,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>v<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo>&sdot;<\/mo><mn>2<\/mn><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">v(t)=\\cos(t^2+5)\\cdot2t<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>v<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>2<\/mn><mi>t<\/mi><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{v(t)=2t\\cos(t^2+5)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Questions<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>1. Which differentiation rule is essential here?<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong> Chain Rule.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>2. What is the inner function?<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">t^2+5<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>3. What is the derivative of the inner function?<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2t<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>4. What is the final velocity function?<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>2<\/mn><mi>t<\/mi><mi>cos<\/mi><mo>&#8289;<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{2t\\cos(t^2+5)}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This case study demonstrates how the Chain Rule converts a nested mathematical expression into a manageable differentiation process. The textbook explicitly introduces this rule for composite functions.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Quiz<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Test your understanding of Exercise 5.2 with this interactive quiz:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Differentiate y = sin(x&sup2; + 5) with respect to x.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1 of 5<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2x sin(x&sup2; + 5)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">x cos(x&sup2; + 5)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2x cos(x&sup2; + 5)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">cos(x&sup2; + 5)Next<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Give feedback<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">CTA<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Master Differentiation with MyMockMate<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Struggling with <strong>Chain Rule, Product Rule, Quotient Rule and Differentiability<\/strong>?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Learn &rarr; Practise &rarr; Test &rarr; Improve<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#128073; <strong>Visit <a href=\"http:\/\/www.mymockmate.com\">www.mymockmate.com<\/a><\/strong> for more NCERT solutions, practice questions, quizzes and exam-focused learning resources.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Keep practising&mdash;every derivative becomes easier when you learn to identify the function layers first.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n<div class=\"mymoc-bottom mymoc-entity-placement\" id=\"mymoc-1577598104\"><div id=\"mymoc-1029778200\"><a href=\"https:\/\/amzn.to\/4xnw9oY\" aria-label=\"head-phones\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones.png 1301w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones-300x80.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones-1024x274.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones-768x205.png 768w\" sizes=\"(max-width: 1301px) 100vw, 1301px\" width=\"1301\" height=\"348\"><\/a><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p>Differentiation Using the Chain Rule | NCERT Class 12 Mathematics Quick Facts dydx=dydu&sdot;dudx\\frac{dy}{dx}=\\frac{dy}{du}\\cdot\\frac{du}{dx} ddx(sin&#8289;x)=cos&#8289;x\\frac{d}{dx}(\\sin x)=\\cos xddx(cos&#8289;x)=&minus;sin&#8289;x\\frac{d}{dx}(\\cos x)=-\\sin xddx(tan&#8289;x)=sec&#8289;2x\\frac{d}{dx}(\\tan x)=\\sec^2xddx(cot&#8289;x)=&minus;csc&#8289;2x\\frac{d}{dx}(\\cot x)=-\\csc^2xddx(sec&#8289;x)=sec&#8289;xtan&#8289;x\\frac{d}{dx}(\\sec x)=\\sec x\\tan xddx(xn)=nxn&minus;1\\frac{d}{dx}(x^n)=nx^{n-1} Story Imagine&#8230;<\/p>\n","protected":false},"author":1,"featured_media":6897,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"postBodyCss":"","postBodyMargin":[],"postBodyPadding":[],"postBodyBackground":{"backgroundType":"classic","gradient":""},"footnotes":""},"categories":[7,8],"tags":[4060,4054,4051,4055,4047,4059,2828,4057,4050,4048,4046,465,4045,4053,4052,4056,4058,308,4049,4018,463,571],"class_list":["post-6896","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-12","category-maths","tag-cbse-class-12","tag-cbse-mathematics","tag-chain-rule","tag-chain-rule-questions","tag-class-12-mathematics","tag-class-12-maths-solutions","tag-continuity-and-differentiability","tag-derivative-problems","tag-derivatives","tag-differentiation","tag-differentiation-solutions","tag-exam-preparation","tag-exercise-5-2","tag-exercise-5-2-solutions","tag-mathematics-solutions","tag-maths-practice","tag-maths-study-material","tag-mymockmate","tag-ncert-class-12","tag-ncert-maths","tag-step-by-step-solutions","tag-www-mymockmate-com"],"featured_image_src":"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/09\/Class-12-Maths-Exercise-5.2-\u2013-Continuity-and-Differentiability.png","author_info":{"display_name":"Team Mymockmate","author_link":"https:\/\/mymockmate.com\/notes\/author\/bsm_adm\/"},"_links":{"self":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/6896","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/comments?post=6896"}],"version-history":[{"count":1,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/6896\/revisions"}],"predecessor-version":[{"id":6898,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/6896\/revisions\/6898"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/6897"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=6896"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=6896"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=6896"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}