Skip to content
← Mathematics

Mathematics · Class 12 Science

Ch 8Differentiation — Class 12 Mathematics, concept-first.

The idea of the derivative goes back to the 17th century, when Sir Isaac Newton and Gottfried Wilhelm Leibniz independently developed the tools of calculus to describe how quantities change.

293

Q&A

10

Concepts

9m

Exam weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Chain Rule

Imagine you're assembling a toy. First you put part A into part B, then you put that combined piece into part C. The final toy's position depends on how you moved A, which then affected B, which then affected C.

Most relevant Q&A

Start with this concept →

In previous exams

How often this chapter’s concepts have been examined — real appearance data, never estimated.

MHT-CET

2 concepts · 46 appearances

2026, 2025, 2024, 2023, 2022, 2021

CA Foundation

1 concept · 1 appearance

2026

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Introduction

The idea of the derivative goes back to the 17th century, when Sir Isaac Newton and Gottfried Wilhelm Leibniz independently developed the tools of calculus to describe how quantities change.

1.1.1

Derivatives of Composite Functions (Function of Another Function)

So far, derivatives have only been found for 'simple' functions — , , — where the standard-function table gives the answer directly.

1.1.2

Theorem: Derivative of a Composite Function (the Chain Rule)

Theorem (Chain Rule). If is a differentiable function of , and is a differentiable function of , then is a differentiable function of , and

1.1.3

Derivatives of Some Standard Composite Functions (Table 1.1.2)

Applying the chain rule to every entry of the Class 11 table, but with a general differentiable inner function in place of plain , produces the composite-function reference table used throughout this…

1.2.1

Geometrical Meaning of Derivative

The derivative also has a purely geometric meaning, quite apart from any algebraic formula. Take a curve and a fixed point on it at , so .

1.2.2

Derivatives of Inverse Functions

If is one-one and onto (so it is invertible), its inverse function exists, and — as the two illustrations below suggest — the derivative of the inverse turns out to be closely tied to the derivative o…

1.2.3

Theorem: Derivative of an Inverse Function

Theorem. If is a differentiable function of with , and exists, then is a differentiable function of , and

1.2.4

Derivatives of Standard Inverse Trigonometric Functions

This section derives the standard derivative of each inverse trigonometric function, by writing as , differentiating implicitly with respect to , and converting the result back into using a Pythagorea…

1.2.5

Table 1.2.1 — Derivatives of Standard Inverse Trigonometric Functions

The six derivatives proved (or set as homework) in the previous section are gathered here into one reference table, each paired with the exact -domain and -range (principal branch) that the formula de…

1.2.6

Derivatives of Standard Inverse Trigonometric Composite Functions, Key Identities and Substitutions

The six inverse-trig derivatives generalise to a composite argument by the chain rule (Table 1.2.2), exactly the way the ordinary trig functions generalised in section 1.1.3 — every formula picks up a…

1.3.1

Logarithmic Differentiation

Some functions are awkward to differentiate directly — long products, quotients or powers of several factors — or are of the genuinely new form , where both the base and the exponent contain ; here ne…

1.3.2

Implicit Functions

Every function met until now has been explicit: (or ) is isolated on one side, written directly as a formula in the other variable — or .

1.3.3

Derivatives of Implicit Functions

Method. (1) Differentiate every term of the equation with respect to , treating throughout as a differentiable function of — so any term containing needs the chain rule (an extra factor of appears eac…

1.4.1

Derivatives of Parametric Functions

Sometimes and are not related to each other directly at all, but each is given as a separate function of a third 'helper' variable , called a parameter: , .

1.4.2

Theorem: Derivative of Parametric Functions

Theorem. If and are differentiable functions of , then is a differentiable function of , and

1.4.3

Differentiation of One Function with Respect to Another Function

If and are both differentiable functions of the same variable , the derivative of with respect to — meaning, treat as if it were the independent variable instead of — is defined as This is exactly the…

1.5.1

Higher Order Derivatives

The derivative of a differentiable function is itself a function of , so — if that new function is itself differentiable — it can be differentiated again, giving what is called the second derivative,…

1.5.2

Successive Differentiation (nth Order Derivative) of Some Standard Functions

There is no single formula that gives the -th derivative of every function — each standard function's own pattern has to be discovered individually.

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 42 questions42 questions
  1. Q1If the function $f(x) = k + x$, for $x < 1$; $= 4x + 3$, for $x \ge 1$ is continuous at $x = 1$ then $k =$ (a) $7$ (b) $8$ (c) $6$ (d) $-6$Preview
  2. Q2If $y = x^x$, find $\dfrac{dy}{dx}$.Preview
  3. Q3If $y = f(u)$ is a differentiable function of $u$ and $u = g(x)$ is a differentiable function of $x$ then prove that $y = f(g(x))$ is a diff…Preview
  4. Q4Discuss the continuity of the following function. If the function has a removable discontinuity, redefine the function so as to remove the d…Preview
  5. Q5If $y = \cos^{-1}\left(2x\sqrt{1 - x^2}\right)$, find $\dfrac{dy}{dx}$Preview
  6. Q6Derivative of $\tan^3\theta$ with respect to $\sec^3\theta$ at $\theta = \dfrac{\pi}{3}$ is (a) $\dfrac{3}{2}$ (b) $\dfrac{\sqrt3}{2}$ (c) $…Preview
  7. Q7Find $\dfrac{dy}{dx}$ if $x\sin y + y\sin x = 0$.Preview
  8. Q8If $f(x) = \dfrac{e^{x^2} - \cos x}{x^2}$, for $x \ne 0$, is continuous at $x = 0$, find $f(0)$.Preview
  9. Q9If $y = f(x)$ is a differentiable function of $x$ such that inverse function $x = f^{-1}(y)$ exists, then prove that $x$ is a differentiable…Preview
  10. Q10Discuss the continuity of the following function, at $x = 0$. $f(x) = \dfrac{x}{|x|}$, for $x \ne 0$; $= 1$, for $x = 0$Preview
  11. Q11If $y = \tan^2(\log x^3)$, find $\dfrac{dy}{dx}$.Preview
  12. Q12If $x = a\cos^3 t$, $y = a\sin^3 t$, show that $\dfrac{dy}{dx} = -\left(\dfrac{y}{x}\right)^{1/3}$.Preview
  13. Q13Examine the continuity of the function: $f(x) = \dfrac{\log 100 + \log(0.01+x)}{3x}$, for $x \neq 0$; $= \dfrac{100}{3}$, for $x = 0$; at $x…Preview
  14. Q14If $f(x) = \dfrac{x^2-9}{x-3} + \alpha$, for $x > 3$; $= 5$, for $x = 3$; $= 2x^2 + 3x + \beta$, for $x < 3$; is continuous at $x = 3$, find…Preview
  15. Q15Find $\dfrac{dy}{dx}$ if $y = \tan^{-1}\left(\dfrac{5x+1}{3-x-6x^2}\right)$.Preview
  16. Q16If $f(x) = (1+2x)^{\frac{1}{x}}$, for $x \neq 0$ is continuous at $x = 0$, then $f(0) =$________. (a) e (b) $e^2$ (c) 0 (d) 2Preview
  17. Q17If $y = x^x$, find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$.Preview
  18. Q18Discuss the continuity of the function $f(x) = \dfrac{\log(2+x)\log(2-x)}{\tan x}$, for $x \neq 0$ $= 1$ for $x = 0$ at the point $x = 0$Preview
  19. Q19If $x = f(t)$ and $y = g(t)$ are differentiable functions of $t$, then prove that $y$ is a differentiable function of $x$ and $\dfrac{\mathr…Preview
  20. Q20If $f(x) = 1 - x$, for $0 < x \le 1$, $= k$, for $x = 0$, is continuous at $x = 0$, then $k = $ ________. (a) 0 (b) $-1$ (c) 2 (d) 1Preview
  21. Q21Differentiate $\sin(x^2+x)$ w.r.t. $x$Preview
  22. Q22Differentiate $\log(\sec x + \tan x)$ w.r.t. $x$.Preview
  23. Q23If $y = x\log x$, then find $\dfrac{d^2y}{dx^2}$.Preview
  24. Q24If $e^x + e^y = e^{x+y}$, show that $\dfrac{dy}{dx} = -e^{y-x}$Preview
  25. Q25Function $f(x)$ is continuous on its domain $[-2, 2]$, where $f(x) = \dfrac{\sin ax}{x}+2$, for $-2 \le x < 0$; $=3x+5$, for $0 \le x \le 1$…Preview
  26. Q26If $f(x) = x^5 + 2x - 3$, then $(f^{-1})'(-3) = $ ________. (a) 0 (b) $-3$ (c) $-\dfrac{1}{3}$ (d) $\dfrac{1}{2}$Preview
  27. Q27If $y = e^{m\tan^{-1}x}$, then show that $(1+x^2)\dfrac{d^2y}{dx^2} + (2x-m)\dfrac{dy}{dx} = 0$Preview
  28. Q28If $x = f(t)$ and $y = g(t)$ are differentiable functions of $t$ so that $y$ is differentiable function of $x$ and $\dfrac{dx}{dt} \ne 0$, t…Preview
  29. Q29If $y$ is a function of $x$ and $\log(x+y) = 2xy$, then the value of $y'(0) = $ ________. (a) 2 (b) 0 (c) $-1$ (d) 1Preview
  30. Q30If $y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + \ldots + \infty}}}$, then show that $\dfrac{dy}{dx} = \dfrac{\sec^2 x}{2y-1}$. Find $\d…Preview
  31. Q31If $y = \cos(m\cos^{-1}x)$ then show that $(1-x^2)\dfrac{d^2y}{dx^2} - x\dfrac{dy}{dx} + m^2y = 0$Preview
  32. Q32The slope of the tangent to the curve $x=\sin\theta$ and $y=\cos 2\theta$ at $\theta = \dfrac{\pi}{6}$ is ____. (a) $-2\sqrt{3}$ (b) $\dfrac…Preview
  33. Q33Find $\dfrac{dy}{dx}$, if $y=(\log x)^x$.Preview
  34. Q34If $y=\sin^{-1}x$, then show that: $(1-x^2)\dfrac{d^2y}{dx^2}-x\cdot\dfrac{dy}{dx}=0$.Preview
  35. Q35If $x=f(t)$ and $y=g(t)$ are differentiable functions of $t$, so that $y$ is function of $x$ and $\dfrac{dx}{dt}\ne 0$ then prove that $\dfr…Preview
  36. Q36Let $f(1)=3$, $f'(1)=-\dfrac{1}{3}$, $g(1)=-4$ and $g'(1)=-\dfrac{8}{3}$. The derivative of $\sqrt{[f(x)]^2+[g(x)]^2}$ w.r.t. $x$ at $x=1$ i…Preview
  37. Q37Find the $n^{th}$ order derivative of $\log x$.Preview
  38. Q38If $x=f(t)$ and $y=g(t)$ are differentiable functions of $t$ so that $y$ is a function of $x$ and if $\dfrac{dx}{dt}\ne 0$ then prove that $…Preview
  39. Q39If $y=\sec(\tan^{-1}x)$, then $\dfrac{dy}{dx}$ at $x=1$ is ____. (a) $\dfrac12$ (b) 1 (c) $\dfrac{1}{\sqrt2}$ (d) $\sqrt2$Preview
  40. Q40Find $\dfrac{dy}{dx}$, if $\sqrt x + \sqrt y = \sqrt a$.Preview
  41. Q41Find $\dfrac{d^2y}{dx^2}$, if $y=x^3+7x^2-2x-9$.Preview
  42. Q42If $y=f(u)$ is a differentiable function of $u$ and $u=g(x)$ is a differentiable function of $x$ then prove that $y$ is a differentiable fun…Preview

More questions

251 Q
+Show 37 questions37 questions
  1. Q1$(x^3-2x-1)^5$Free
  2. Q2$\left(2x^{3/2}-3x^{4/3}-5\right)^{5/2}$Free
  3. Q3$\sqrt{x^2+4x-7}$Free
  4. Q4$\sqrt{x^2+\sqrt{x^2+1}}$Preview
  5. Q5$\dfrac{8}{3\sqrt[3]{(2x^2-7x-5)^{11}}}$Preview
  6. Q6$\left(\sqrt{3x-5}-\dfrac{1}{\sqrt{3x-5}}\right)^5$Preview
  7. Q7$\cos(x^2+a^2)$Preview
  8. Q8$\sqrt{e^{(3x+2)}+5}$Preview
  9. Q9$\log\left[\tan\left(\dfrac{x}{2}\right)\right]$Preview
  10. Q10$\sqrt{\tan\sqrt{x}}$Preview
  11. Q11$\cot^3[\log(x^3)]$Preview
  12. Q12$5^{\sin^3 x + 3}$Preview
  13. Q13$\text{cosec}(\sqrt{\cos x})$Preview
  14. Q14$\log[\cos(x^3-5)]$Preview
  15. Q15$e^{3\sin^2 x - 2\cos^2 x}$Preview
  16. Q16$\cos^2[\log(x^2+7)]$Preview
  17. Q17$\tan[\cos(\sin x)]$Preview
  18. Q18$\sec[\tan(x^4+4)]$Preview
  19. Q19$e^{\log[(\log x)^2 - \log(x^2)]}$Preview
  20. Q20$\sin\sqrt{\sin\sqrt{x}}$Preview
  21. Q21$\log[\sec(e^{x^2})]$Preview
  22. Q22$\log_{e^2}(\log x)$Preview
  23. Q23$\{\log[\log(\log x)]\}^2$Preview
  24. Q24$\sin^2(x^2) - \cos^2(x^2)$Preview
  25. Q25$(x^2+4x+1)^3 + (x^3-5x-2)^4$Preview
  26. Q26$(1+4x)^5(3+x-x^2)^8$Preview
  27. Q27$\dfrac{x}{\sqrt{7-3x}}$Preview
  28. Q28$\dfrac{(x^3-5)^5}{(x^3+3)^3}$Preview
  29. Q29$(1+\sin^2 x)^2(1+\cos^2 x)^3$Preview
  30. Q30$\sqrt{\cos x} + \sqrt{\cos\sqrt{x}}$Preview
  31. Q31$\log(\sec 3x + \tan 3x)$Preview
  32. Q32$\log[\tan^3 x \cdot \sin^4 x \cdot (x^2+7)^7]$Preview
  33. Q33$(25)^{\log_5(\sec x)} - (16)^{\log_4(\tan x)}$Preview
  34. Q34A table of values of f, g, f' and g' is given: at $x=2$: $f(x)=1$, $g(x)=6$, $f'(x)=-3$, $g'(x)=4$; at $x=4$: $f(x)=3$, $g(x)=4$, $f'(x)=5$,…Preview
  35. Q35If $f'(3) = -1$, $g'(2) = 5$, $g(2) = 3$ and $y = f[g(x)]$, find $\left(\dfrac{dy}{dx}\right)_{x=2}$.Preview
  36. Q36Find the x co-ordinates of all the points on the curve $y = \sin 2x - 2\sin x$, $0 \le x < 2\pi$, where $\dfrac{dy}{dx} = 0$.Preview
  37. Q37Select the appropriate hint from the hint basket and fill in the blank spaces in the following paragraph. [Activity] "Let $f(x) = x^2 + 5$ a…Preview
+Show 69 questions69 questions
  1. Q38Find the derivative of the function $y=f(x)$ using the derivative of the inverse function $x=f^{-1}(y)$: $y=\sqrt{x}$Free
  2. Q39Find the derivative of the function $y=f(x)$ using the derivative of the inverse function $x=f^{-1}(y)$: $y=2-\sqrt{x}$Free
  3. Q40Find the derivative of the function $y=f(x)$ using the derivative of the inverse function $x=f^{-1}(y)$: $y=\sqrt[3]{x-2}$Free
  4. Q41Find the derivative of the function $y=f(x)$ using the derivative of the inverse function $x=f^{-1}(y)$: $y=\log(2x-1)$Preview
  5. Q42Find the derivative of the function $y=f(x)$ using the derivative of the inverse function $x=f^{-1}(y)$: $y=2x+3$Preview
  6. Q43Find the derivative of the function $y=f(x)$ using the derivative of the inverse function $x=f^{-1}(y)$: $y=e^{x-3}$Preview
  7. Q44Find the derivative of the function $y=f(x)$ using the derivative of the inverse function $x=f^{-1}(y)$: $y=e^{2x-3}$Preview
  8. Q45Find the derivative of the function $y=f(x)$ using the derivative of the inverse function $x=f^{-1}(y)$: $y=\log_2\left(\dfrac{x}{2} ight)$Preview
  9. Q46Find the derivative of the inverse function of the following: $y=x^2 e^x$Preview
  10. Q47Find the derivative of the inverse function of the following: $y=x\cos x$Preview
  11. Q48Find the derivative of the inverse function of the following: $y=x\cdot 7^x$Preview
  12. Q49Find the derivative of the inverse function of the following: $y=x^2+\log x$Preview
  13. Q50Find the derivative of the inverse function of the following: $y=x\log x$Preview
  14. Q51Find the derivative of the inverse of the following function, and also find its value at the point indicated: $y=x^5+2x^3+3x$, at $x=1$Preview
  15. Q52Find the derivative of the inverse of the following function, and also find its value at the point indicated: $y=e^x+3x+2$, at $x=0$Preview
  16. Q53Find the derivative of the inverse of the following function, and also find its value at the point indicated: $y=3x^2+2\log x^3$, at $x=1$Preview
  17. Q54Find the derivative of the inverse of the following function, and also find its value at the point indicated: $y=\sin(x-2)+x^2$, at $x=2$Preview
  18. Q55If $f(x)=x^3+x-2$, find $(f^{-1})'(0)$.Preview
  19. Q56Using derivative, prove: $\tan^{-1}x+\cot^{-1}x=\dfrac{\pi}{2}$Preview
  20. Q57Using derivative, prove: $\sec^{-1}x+\text{cosec}^{-1}x=\dfrac{\pi}{2}$, for $|x|\ge1$Preview
  21. Q58Differentiate the following w.r.t. $x$: $\tan^{-1}(\log x)$Preview
  22. Q59Differentiate the following w.r.t. $x$: $\text{cosec}^{-1}(e^{-x})$Preview
  23. Q60Differentiate the following w.r.t. $x$: $\cot^{-1}(x^3)$Preview
  24. Q61Differentiate the following w.r.t. $x$: $\cot^{-1}(4^x)$Preview
  25. Q62Differentiate the following w.r.t. $x$: $\tan^{-1}(\sqrt{x})$Preview
  26. Q63Differentiate the following w.r.t. $x$: $\sin^{-1}\left(\dfrac{1+x^2}{2}\right)$Preview
  27. Q64Differentiate the following w.r.t. $x$: $\cos^{-1}(1-x^2)$Preview
  28. Q65Differentiate the following w.r.t. $x$: $\sin^{-1}(x^{3/2})$Preview
  29. Q66Differentiate the following w.r.t. $x$: $\cos^3[\cos^{-1}(x^3)]$Preview
  30. Q67Differentiate the following w.r.t. $x$: $\sin^4[\sin^{-1}(\sqrt{x})]$Preview
  31. Q68Differentiate the following w.r.t. $x$: $\cot^{-1}[\cot(e^{x^2})]$Preview
  32. Q69Differentiate the following w.r.t. $x$: $\text{cosec}^{-1}\left(\dfrac{1}{\cos(5x)}\right)$Preview
  33. Q70Differentiate the following w.r.t. $x$: $\cos^{-1}\sqrt{\dfrac{1+\cos x}{2}}$Preview
  34. Q71Differentiate the following w.r.t. $x$: $\cos^{-1}\sqrt{\dfrac{1-\cos(x^2)}{2}}$Preview
  35. Q72Differentiate the following w.r.t. $x$: $\tan^{-1}\dfrac{1-\tan(x/2)}{1+\tan(x/2)}$Preview
  36. Q73Differentiate the following w.r.t. $x$: $\text{cosec}^{-1}\left(\dfrac{1}{4\cos^3 2x - 3\cos 2x}\right)$Preview
  37. Q74Differentiate the following w.r.t. $x$: $\tan^{-1}\dfrac{1+\cos(x/3)}{\sin(x/3)}$Preview
  38. Q75Differentiate the following w.r.t. $x$: $\cot^{-1}\dfrac{\sin 3x}{1+\cos 3x}$Preview
  39. Q76Differentiate the following w.r.t. $x$: $\tan^{-1}\dfrac{\cos 7x}{1+\sin 7x}$Preview
  40. Q77Differentiate the following w.r.t. $x$: $\tan^{-1}\sqrt{\dfrac{1+\cos x}{1-\cos x}}$Preview
  41. Q78Differentiate the following w.r.t. $x$: $\tan^{-1}(\text{cosec}\ x + \cot x)$Preview
  42. Q79Differentiate the following w.r.t. $x$: $\cot^{-1}\dfrac{\sqrt{1+\sin(4x/3)}+\sqrt{1-\sin(4x/3)}}{\sqrt{1+\sin(4x/3)}-\sqrt{1-\sin(4x/3)}}$Preview
  43. Q80Differentiate the following w.r.t. $x$: $\sin^{-1}\dfrac{4\sin x+5\cos x}{\sqrt{41}}$Preview
  44. Q81Differentiate the following w.r.t. $x$: $\cos^{-1}\dfrac{\sqrt3\cos x-\sin x}{2}$Preview
  45. Q82Differentiate the following w.r.t. $x$: $\sin^{-1}\dfrac{\cos\sqrt x+\sin\sqrt x}{\sqrt2}$Preview
  46. Q83Differentiate the following w.r.t. $x$: $\cos^{-1}\dfrac{3\cos 3x-4\sin 3x}{5}$Preview
  47. Q84Differentiate the following w.r.t. $x$: $\cos^{-1}\dfrac{3\cos(e^x)+2\sin(e^x)}{\sqrt{13}}$Preview
  48. Q85Differentiate the following w.r.t. $x$: $\text{cosec}^{-1}\dfrac{10}{6\sin(2x)-8\cos(2x)}$Preview
  49. Q86Differentiate the following w.r.t. $x$: $\cos^{-1}\dfrac{1-x^2}{1+x^2}$Preview
  50. Q87Differentiate the following w.r.t. $x$: $\tan^{-1}\dfrac{2x}{1-x^2}$Preview
  51. Q88Differentiate the following w.r.t. $x$: $\sin^{-1}\dfrac{1-x^2}{1+x^2}$Preview
  52. Q89Differentiate the following w.r.t. $x$: $\sin^{-1}(2x\sqrt{1-x^2})$Preview
  53. Q90Differentiate the following w.r.t. $x$: $\cos^{-1}(3x-4x^3)$Preview
  54. Q91Differentiate the following w.r.t. $x$: $\cos^{-1}\dfrac{e^x-e^{-x}}{e^x+e^{-x}}$Preview
  55. Q92Differentiate the following w.r.t. $x$: $\cos^{-1}\dfrac{1-9^x}{1+9^x}$Preview
  56. Q93Differentiate the following w.r.t. $x$: $\sin^{-1}\dfrac{4^{x+\frac12}}{1+2^{4x}}$Preview
  57. Q94Differentiate the following w.r.t. $x$: $\sin^{-1}\dfrac{1-25x^2}{1+25x^2}$Preview
  58. Q95Differentiate the following w.r.t. $x$: $\sin^{-1}\dfrac{1-x^3}{1+x^3}$Preview
  59. Q96Differentiate the following w.r.t. $x$: $\tan^{-1}\dfrac{2x^{5/2}}{1-x^5}$Preview
  60. Q97Differentiate the following w.r.t. $x$: $\cot^{-1}\dfrac{1-\sqrt x}{1+\sqrt x}$Preview
  61. Q98Differentiate the following w.r.t. $x$: $\tan^{-1}\dfrac{8x}{1-15x^2}$Preview
  62. Q99Differentiate the following w.r.t. $x$: $\cot^{-1}\dfrac{1+35x^2}{2x}$Preview
  63. Q100Differentiate the following w.r.t. $x$: $\tan^{-1}\dfrac{2\sqrt x}{1+3x}$Preview
  64. Q101Differentiate the following w.r.t. $x$: $\tan^{-1}\dfrac{2^{x+1}}{1-3(4^x)}$Preview
  65. Q102Differentiate the following w.r.t. $x$: $\tan^{-1}\dfrac{2^x}{1+2^{2x+1}}$Preview
  66. Q103Differentiate the following w.r.t. $x$: $\cot^{-1}\dfrac{a^2-6x^2}{5ax}$Preview
  67. Q104Differentiate the following w.r.t. $x$: $\tan^{-1}\dfrac{a+b\tan x}{b-a\tan x}$Preview
  68. Q105Differentiate the following w.r.t. $x$: $\tan^{-1}\dfrac{5-x}{6x^2-5x-3}$Preview
  69. Q106Differentiate the following w.r.t. $x$: $\cot^{-1}\dfrac{4-x-2x^2}{3x+2}$Preview
+Show 44 questions44 questions
  1. Q107Differentiate the following w.r.t. $x$: $\dfrac{(x+1)^2}{(x+2)^3(x+3)^4}$Free
  2. Q108Differentiate the following w.r.t. $x$: $\dfrac{4x-1}{(2x+3)(5-2x)^2}$Free
  3. Q109Differentiate the following w.r.t. $x$: $(x^2+3)^{3/2}\cdot\sin^3(2x)\cdot 2^{x^2}$Free
  4. Q110Differentiate the following w.r.t. $x$: $\dfrac{(x^2+2x+2)^{3/2}}{(\sqrt x+3)^3(\cos x)^x}$Preview
  5. Q111Differentiate the following w.r.t. $x$: $\dfrac{x^5\tan^3 4x}{\sin^2 3x}$Preview
  6. Q112Differentiate the following w.r.t. $x$: $x^{\tan^{-1}x}$Preview
  7. Q113Differentiate the following w.r.t. $x$: $(\sin x)^x$Preview
  8. Q114Differentiate the following w.r.t. $x$: $(\sin x)^{x^3}$Preview
  9. Q115Differentiate the following w.r.t. $x$: $x^e+x^x+e^x+e^e$Preview
  10. Q116Differentiate the following w.r.t. $x$: $x^{x^x}+e^{x^x}$Preview
  11. Q117Differentiate the following w.r.t. $x$: $(\log x)^x-(\cos x)^{\cot x}$Preview
  12. Q118Differentiate the following w.r.t. $x$: $x^{e^x}+(\log x)^{\sin x}$Preview
  13. Q119Differentiate the following w.r.t. $x$: $e^{\tan x}+(\log x)^{\tan x}$Preview
  14. Q120Differentiate the following w.r.t. $x$: $(\sin x)^{\tan x}+(\cos x)^{\cot x}$Preview
  15. Q121Differentiate the following w.r.t. $x$: $10^{x^x}+x^{x^{10}}+x^{10^x}$Preview
  16. Q122Differentiate the following w.r.t. $x$: $[(\tan x)^{\tan x}]^{\tan x}$ at $x=\dfrac{\pi}{4}$Preview
  17. Q123Find $\dfrac{dy}{dx}$ if $\sqrt x+\sqrt y=\sqrt a$Preview
  18. Q124Find $\dfrac{dy}{dx}$ if $x\sqrt x+y\sqrt y=a\sqrt a$Preview
  19. Q125Find $\dfrac{dy}{dx}$ if $x+\sqrt{xy}+y=1$Preview
  20. Q126Find $\dfrac{dy}{dx}$ if $x^3+x^2y+xy^2+y^3=81$Preview
  21. Q127Find $\dfrac{dy}{dx}$ if $x^2y^2-\tan^{-1}\sqrt{x^2+y^2}=\cot^{-1}\sqrt{x^2+y^2}$Preview
  22. Q128Find $\dfrac{dy}{dx}$ if $xe^y+ye^x=1$Preview
  23. Q129Find $\dfrac{dy}{dx}$ if $e^{x+y}=\cos(x-y)$Preview
  24. Q130Find $\dfrac{dy}{dx}$ if $\cos(xy)=x+y$Preview
  25. Q131Find $\dfrac{dy}{dx}$ if $e^{e^{x-y}}=\dfrac{x}{y}$Preview
  26. Q132Find $\dfrac{dy}{dx}$ if $x+\sin(x+y)=y-\cos(x-y)$Preview
  27. Q133Show that $\dfrac{dy}{dx}=\dfrac{y}{x}$ in the following, where $a$ and $p$ are constants: $x^7y^5=(x+y)^{12}$Preview
  28. Q134Show that $\dfrac{dy}{dx}=\dfrac{y}{x}$ in the following, where $a$ and $p$ are constants: $x^py^4=(x+y)^{p+4}$, $p\in N$Preview
  29. Q135Show that $\dfrac{dy}{dx}=\dfrac{y}{x}$ in the following, where $a$ and $p$ are constants: $\sec\dfrac{x^5+y^5}{x^5-y^5}=a^2$Preview
  30. Q136Show that $\dfrac{dy}{dx}=\dfrac{y}{x}$ in the following, where $a$ and $p$ are constants: $\tan^{-1}\dfrac{3x^2-4y^2}{3x^2+4y^2}=a^2$Preview
  31. Q137Show that $\dfrac{dy}{dx}=\dfrac{y}{x}$ in the following, where $a$ and $p$ are constants: $\cos^{-1}\dfrac{7x^4+5y^4}{7x^4-5y^4}=\tan^{-1}a…Preview
  32. Q138Show that $\dfrac{dy}{dx}=\dfrac{y}{x}$ in the following, where $a$ and $p$ are constants: $\log\dfrac{x^{20}-y^{20}}{x^{20}+y^{20}}=20$Preview
  33. Q139Show that $\dfrac{dy}{dx}=\dfrac{y}{x}$ in the following, where $a$ and $p$ are constants: $e^{\frac{x^7-y^7}{x^7+y^7}}=a$Preview
  34. Q140Show that $\dfrac{dy}{dx}=\dfrac{y}{x}$ in the following, where $a$ and $p$ are constants: $\sin\dfrac{x^3-y^3}{x^3+y^3}=a^3$Preview
  35. Q141If $\log(x+y)=\log(xy)+p$ ($p$ constant), prove $\dfrac{dy}{dx}=-\dfrac{y^2}{x^2}$.Preview
  36. Q142If $\log_{10}\dfrac{x^3-y^3}{x^3+y^3}=2$, show $\dfrac{dy}{dx}=-\dfrac{99x^2}{101y^2}$.Preview
  37. Q143If $\log_5\dfrac{x^4+y^4}{x^4-y^4}=2$, show $\dfrac{dy}{dx}=-\dfrac{12x^3}{13y^3}$.Preview
  38. Q144If $e^x+e^y=e^{x+y}$, show $\dfrac{dy}{dx}=-e^{y-x}$.Preview
  39. Q145If $\sin^{-1}\dfrac{x^5-y^5}{x^5+y^5}=\dfrac{\pi}{6}$, show $\dfrac{dy}{dx}=\dfrac{x^4}{3y^4}$.Preview
  40. Q146If $x^y=e^{x-y}$, show $\dfrac{dy}{dx}=\dfrac{\log x}{(1+\log x)^2}$.Preview
  41. Q147If $y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\cdots\infty}}}$, show $\dfrac{dy}{dx}=\dfrac{\sin x}{1-2y}$.Preview
  42. Q148If $y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+\cdots\infty}}}$, show $\dfrac{dy}{dx}=\dfrac{1}{x(2y-1)}$.Preview
  43. Q149If $y=x^{x^{x^{\cdots\infty}}}$, show $\dfrac{dy}{dx}=\dfrac{y^2}{x(1-\log y)}$.Preview
  44. Q150If $e^y=y^x$, show $\dfrac{dy}{dx}=\dfrac{(\log y)^2}{\log y-1}$.Preview
+Show 29 questions29 questions
  1. Q151Find $\dfrac{dy}{dx}$ if $x=at^2$, $y=2at$.Free
  2. Q152Find $\dfrac{dy}{dx}$ if $x=a\cot\theta$, $y=b\,\text{cosec}\,\theta$.Free
  3. Q153Find $\dfrac{dy}{dx}$ if $x=\sqrt{a^2+m^2}$, $y=\log(a^2+m^2)$ (parameter $m$).Free
  4. Q154Find $\dfrac{dy}{dx}$ if $x=\sin\theta$, $y=\tan\theta$.Preview
  5. Q155Find $\dfrac{dy}{dx}$ if $x=a(1-\cos\theta)$, $y=b(\theta-\sin\theta)$.Preview
  6. Q156Find $\dfrac{dy}{dx}$ if $x=\left(t+\dfrac1t\right)^a$, $y=at+\dfrac1t$, where $a>0,\ a\ne1,\ t\ne0$.Preview
  7. Q157Find $\dfrac{dy}{dx}$ if $x=\cos^{-1}\dfrac{2t}{1+t^2}$, $y=\sec^{-1}\left(\sqrt{1+t^2}\right)$.Preview
  8. Q158Find $\dfrac{dy}{dx}$ if $x=\cos^{-1}(4t^3-3t)$, $y=\tan^{-1}\dfrac{\sqrt{1-t^2}}{t}$.Preview
  9. Q159Find $\dfrac{dy}{dx}$ if $x=\text{cosec}^2\theta$, $y=\cot^3\theta$, at $\theta=\dfrac{\pi}{6}$.Preview
  10. Q160Find $\dfrac{dy}{dx}$ if $x=a\cos^3\theta$, $y=a\sin^3\theta$, at $\theta=\dfrac{\pi}{3}$.Preview
  11. Q161Find $\dfrac{dy}{dx}$ if $x=t^2+t+1$, $y=\sin\dfrac{\pi t}{2}+\cos\dfrac{\pi t}{2}$, at $t=1$.Preview
  12. Q162Find $\dfrac{dy}{dx}$ if $x=2\cos t+\cos 2t$, $y=2\sin t-\sin 2t$, at $t=\dfrac{\pi}{4}$.Preview
  13. Q163Find $\dfrac{dy}{dx}$ if $x=t+2\sin(\pi t)$, $y=3t-\cos(\pi t)$, at $t=\dfrac12$.Preview
  14. Q164If $x=a\sqrt{\sec\theta-\tan\theta}$, $y=a\sqrt{\sec\theta+\tan\theta}$, show that $\dfrac{dy}{dx}=-\dfrac{y}{x}$.Preview
  15. Q165If $x=e^{\sin3t}$, $y=e^{\cos3t}$, show that $\dfrac{dy}{dx}=-\dfrac{y\log x}{x\log y}$.Preview
  16. Q166If $x=\dfrac{t+1}{t-1}$, $y=\dfrac{t-1}{t+1}$, show that $y^2+\dfrac{dy}{dx}=0$.Preview
  17. Q167If $x=a\cos^3t$, $y=a\sin^3t$, show that $\dfrac{dy}{dx}=-\left(\dfrac{y}{x}\right)^{1/3}$.Preview
  18. Q168If $x=2\cos^4(t+3)$, $y=3\sin^4(t+3)$, show that $\dfrac{dy}{dx}=-\sqrt{\dfrac{3y}{2x}}$.Preview
  19. Q169If $x=\log(1+t^2)$, $y=t-\tan^{-1}t$, show that $\dfrac{dy}{dx}=\dfrac{\sqrt{e^x-1}}{2}$.Preview
  20. Q170If $x=\sin^{-1}(e^t)$, $y=\sqrt{1-e^{2t}}$, show that $\sin x+\dfrac{dy}{dx}=0$.Preview
  21. Q171If $x=\dfrac{2bt}{1+t^2}$, $y=a\dfrac{1-t^2}{1+t^2}$, show that $\dfrac{dx}{dy}=-\dfrac{b^2y}{a^2x}$.Preview
  22. Q172Differentiate $x\sin x$ w.r.t. $\tan x$.Preview
  23. Q173Differentiate $\sin^{-1}\dfrac{2x}{1+x^2}$ w.r.t. $\cos^{-1}\dfrac{1-x^2}{1+x^2}$.Preview
  24. Q174Differentiate $\tan^{-1}\dfrac{x}{\sqrt{1-x^2}}$ w.r.t. $\sec^{-1}\dfrac{1}{2x^2-1}$.Preview
  25. Q175Differentiate $\cos^{-1}\dfrac{1-x^2}{1+x^2}$ w.r.t. $\tan^{-1}x$.Preview
  26. Q176Differentiate $3^x$ w.r.t. $\log_x 3$.Preview
  27. Q177Differentiate $\tan^{-1}\dfrac{\cos x}{1+\sin x}$ w.r.t. $\sec^{-1}x$.Preview
  28. Q178Differentiate $x^x$ w.r.t. $x^{\sin x}$.Preview
  29. Q179Differentiate $\tan^{-1}\dfrac{\sqrt{1+x^2}-1}{x}$ w.r.t. $\tan^{-1}\dfrac{2x\sqrt{1-x^2}}{1-2x^2}$.Preview
+Show 34 questions34 questions
  1. Q180$2x^5-4x^3-\dfrac{2}{x^2}-9$Free
  2. Q181$e^{2x}\cdot\tan x$Free
  3. Q182$e^{4x}\cdot\cos 5x$Free
  4. Q183$x^3\log x$Preview
  5. Q184$\log(\log x)$Preview
  6. Q185$x^x$Preview
  7. Q186$x=a(\theta-\sin\theta)$, $y=a(1-\cos\theta)$Preview
  8. Q187$x=2at^2$, $y=4at$Preview
  9. Q188$x=\sin\theta$, $y=\sin^3\theta$, when $\theta=\pi/2$Preview
  10. Q189$x=a\cos\theta$, $y=b\sin\theta$, at $\theta=\pi/4$Preview
  11. Q190$x=at^2$, $y=2at$, then show that $xy\,\dfrac{d^2y}{dx^2}+a=0$Preview
  12. Q191$y=e^{m\tan^{-1}x}$, show that $(1+x^2)\dfrac{d^2y}{dx^2}+(2x-m)\dfrac{dy}{dx}=0$Preview
  13. Q192$x=\cos t$, $y=e^{mt}$, show that $(1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}-m^2y=0$Preview
  14. Q193$y=x+\tan x$, show that $\cos^2x\cdot\dfrac{d^2y}{dx^2}-2y+2x=0$Preview
  15. Q194$y=e^{ax}\sin(bx)$, show that $y_2-2ay_1+(a^2+b^2)y=0$ (where $y_1=dy/dx$, $y_2=d^2y/dx^2$)Preview
  16. Q195$\sec^{-1}\dfrac{7x^3-5y^3}{7x^3+5y^3}=m$, show that $\dfrac{d^2y}{dx^2}=0$Preview
  17. Q196$2y=\sqrt{x+1}+\sqrt{x-1}$, show that $4(x^2-1)y_2+4xy_1-y=0$Preview
  18. Q197$y=[\log(x+\sqrt{x^2+a^2})]^m$, show that $(x^2+a^2)\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}=0$Preview
  19. Q198$y=\sin(m\cos^{-1}x)$, show that $(1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}+m^2y=0$Preview
  20. Q199$y=\log(\log 2x)$, show that $xy_2+y_1(1+xy_1)=0$Preview
  21. Q200$x^2+6xy+y^2=10$, show that $\dfrac{d^2y}{dx^2}=\dfrac{80}{(3x+y)^3}$Preview
  22. Q201$x=a\sin t-b\cos t$, $y=a\cos t+b\sin t$, show that $\dfrac{d^2y}{dx^2}=-\dfrac{x^2+y^2}{y^3}$Preview
  23. Q202$(ax+b)^m$Preview
  24. Q203$\dfrac1x$Preview
  25. Q204$e^{ax+b}$Preview
  26. Q205$a^{px+q}$Preview
  27. Q206$\log(ax+b)$Preview
  28. Q207$\cos x$Preview
  29. Q208$\sin(ax+b)$Preview
  30. Q209$\cos(3-2x)$Preview
  31. Q210$\log(2x+3)$Preview
  32. Q211$\dfrac{1}{3x-5}$Preview
  33. Q212$y=e^{ax}\cos(bx+c)$Preview
  34. Q213$y=e^{8x}\cos(6x+7)$Preview
+Show 12 questions12 questions
  1. Q214Let $f(1)=3$, $f'(1)=-\dfrac13$, $g(1)=-4$, $g'(1)=-\dfrac83$. The derivative of $\sqrt{[f(x)]^2+[g(x)]^2}$ w.r.t. x at $x=1$ is: (A) $-\dfr…Free
  2. Q215If $y=\sec(\tan^{-1}x)$ then $\dfrac{dy}{dx}$ at $x=1$ is equal to: (A) $\dfrac12$ (B) $1$ (C) $\dfrac{1}{\sqrt2}$ (D) $\sqrt2$Free
  3. Q216If $f(x)=\sin^{-1}\dfrac{4^{x+\frac12}}{1+2^{4x}}$, which of the following is NOT the derivative of $f(x)$? (A) $\dfrac{2\cdot4^x\log4}{1+4^…Free
  4. Q217If $x^y=y^x$, then $\dfrac{dy}{dx}=$ (A) $\dfrac{x(x\log y-y)}{y(y\log x-x)}$ (B) $\dfrac{y(y\log x-x)}{x(x\log y-y)}$ (C) $\dfrac{y^2(1-\lo…Preview
  5. Q218If $y=\sin(2\sin^{-1}x)$, then $\dfrac{dy}{dx}=$ (A) $\dfrac{2-4x^2}{\sqrt{1-x^2}}$ (B) $\dfrac{2+4x^2}{\sqrt{1-x^2}}$ (C) $\dfrac{4x^2-1}{\…Preview
  6. Q219If $y=\tan^{-1}\dfrac{x}{1+\sqrt{1-x^2}}+\sin\!\left(2\tan^{-1}\sqrt{\dfrac{1-x}{1+x}}\right)$, then $\dfrac{dy}{dx}=$ (A) $\dfrac{x}{\sqrt{…Preview
  7. Q220If y is a function of x and $\log(x+y)=2xy$, then the value of $y'(0)$ = (A) 2 (B) 0 (C) $-1$ (D) 1Preview
  8. Q221If g is the inverse of a function f and $f'(x)=\dfrac{1}{1+x^7}$, then the value of $g'(x)$ is equal to: (A) $1+x^7$ (B) $\dfrac{1}{1+[g(x)]…Preview
  9. Q222If $x\sqrt{y+1}+y\sqrt{x+1}=0$ and $x\ne y$ then $\dfrac{dy}{dx}=$ (A) $\dfrac{1}{(1+x)^2}$ (B) $-\dfrac{1}{(1+x)^2}$ (C) $(1+x)^2$ (D) $-\d…Preview
  10. Q223If $y=\tan^{-1}\sqrt{\dfrac{a-x}{a+x}}$, where $-a<x<a$ then $\dfrac{dy}{dx}=$ (A) $\dfrac{x}{\sqrt{a^2-x^2}}$ (B) $\dfrac{a}{\sqrt{a^2-x^2}…Preview
  11. Q224If $x=a(\cos\theta+\theta\sin\theta)$, $y=a(\sin\theta-\theta\cos\theta)$ then $\left(\dfrac{d^2y}{dx^2}\right)_{\theta=\pi/4}=$ (A) $\dfrac…Preview
  12. Q225If $y=a\cos(\log x)$ and $A\dfrac{d^2y}{dx^2}+B\dfrac{dy}{dx}+Cy=0$, then the values of A, B, C are: (A) $x^2,-x,-1$ (B) $x^2,x,1$ (C) $x^2,…Preview
+Show 26 questions26 questions
  1. Q226Let $f(x) = -x$ for $-2\le x<0$; $f(x)=2x$ for $0\le x\le2$; $f(x)=\dfrac{2x-4}{3}$ for $2<x\le7$. And $g(x)=6-3x$ for $0\le x\le2$; $g(x)=\…Free
  2. Q227The values of $f(x)$, $g(x)$, $f'(x)$ and $g'(x)$ are given in the following table: at $x=-1$: $f=3,g=2,f'=-3,g'=4$; at $x=2$: $f=2,g=-1,f'=…Free
  3. Q228Suppose that the functions f and g and their derivatives with respect to x have the following values at $x=0$ and $x=1$: at $x=0$: $f=1,g=1,…Free
  4. Q229Suppose that the functions f and g and their derivatives with respect to x have the following values at $x=0$ and $x=1$: at $x=0$: $f=1,g=1,…Preview
  5. Q230Suppose that the functions f and g and their derivatives with respect to x have the following values at $x=0$ and $x=1$: at $x=0$: $f=1,g=1,…Preview
  6. Q231Suppose that the functions f and g and their derivatives with respect to x have the following values at $x=0$ and $x=1$: at $x=0$: $f=1,g=1,…Preview
  7. Q232Differentiate $\sin\!\left[2\tan^{-1}\dfrac{1-x}{1+x}\right]$ w.r.t. xPreview
  8. Q233Differentiate $\sin^2\!\left[\cot^{-1}\dfrac{1+x}{1-x}\right]$ w.r.t. xPreview
  9. Q234Differentiate $\tan^{-1}\dfrac{\sqrt{x(3-x)}}{1-3x}$ w.r.t. xPreview
  10. Q235Differentiate $\cos^{-1}\dfrac{\sqrt{1+x}-\sqrt{1-x}}{2}$ w.r.t. xPreview
  11. Q236Differentiate $\tan^{-1}\dfrac{x}{1+6x^2}+\cot^{-1}\dfrac{1-10x^2}{7x}$ w.r.t. xPreview
  12. Q237Differentiate $\tan^{-1}\dfrac{\sqrt{1+x^2}+x}{\sqrt{1+x^2}-x}$ w.r.t. xPreview
  13. Q238If $\sqrt{y+x}+\sqrt{y-x}=c$, then show that $\dfrac{dy}{dx}=\dfrac{y}{x}-\sqrt{\dfrac{y^2}{x^2}-1}$Preview
  14. Q239If $x\sqrt{1-y^2}+y\sqrt{1-x^2}=1$, then show that $\dfrac{dy}{dx}=-\sqrt{\dfrac{1-y^2}{1-x^2}}$Preview
  15. Q240If $x\sin(a+y)+\sin a\cos(a+y)=0$, then show that $\dfrac{dy}{dx}=\dfrac{\sin^2(a+y)}{\sin a}$Preview
  16. Q241If $\sin y=x\sin(a+y)$, then show that $\dfrac{dy}{dx}=\dfrac{\sin^2(a+y)}{\sin a}$Preview
  17. Q242If $x=e^{x/y}$, then show that $\dfrac{dy}{dx}=\dfrac{x-y}{x\log x}$Preview
  18. Q243If $y=f(x)$ is a differentiable function then show that $\dfrac{d^2x}{dy^2}=-\left(\dfrac{dy}{dx}\right)^{-3}\cdot\dfrac{d^2y}{dx^2}$Preview
  19. Q244Differentiate $\tan^{-1}\dfrac{\sqrt{1+x^2}-1}{x}$ w.r.t. $\tan^{-1}\dfrac{2x\sqrt{1-x^2}}{1-2x^2}$Preview
  20. Q245Differentiate $\log\dfrac{\sqrt{1+x^2}+x}{\sqrt{1+x^2}-x}$ w.r.t. $\cos(\log x)$Preview
  21. Q246Differentiate $\tan^{-1}\dfrac{\sqrt{1+x^2}-1}{x}$ w.r.t. $\cos^{-1}\dfrac{1+\sqrt{1+x^2}}{2\sqrt{1+x^2}}$Preview
  22. Q247If $y^2=a^2\cos^2x+b^2\sin^2x$, show that $y+\dfrac{d^2y}{dx^2}=\dfrac{a^2b^2}{y^3}$Preview
  23. Q248If $\log y=\log(\sin x)-x^2$, show that $\dfrac{d^2y}{dx^2}+4x\dfrac{dy}{dx}+(4x^2+3)y=0$Preview
  24. Q249If $x=a\cos\theta$, $y=b\sin\theta$, show that $a^2y\dfrac{d^2y}{dx^2}+\left(\dfrac{dy}{dx}\right)^2+b^2=0$Preview
  25. Q250If $y=A\cos(\log x)+B\sin(\log x)$, show that $x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+y=0$Preview
  26. Q251If $y=Ae^{mx}+Be^{nx}$, show that $\dfrac{d^2y}{dx^2}-(m+n)\dfrac{dy}{dx}+mn\,y=0$Preview