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Mathematics · Class 11 Science

Ch 10Differential Calculus – Differentiability and Methods of Differentiation — Class 11 Mathematics, concept-first.

Everyone has an intuitive sense of speed as "distance covered per unit time," but that intuition is really about average speed. If a bus covers km in one hour, its average velocity for the trip is km/h — yet it plainly slows in towns and speeds up on open stretches, so the actual velocity keeps changing.

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Key concepts

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The Derivative — Definition

The derivative grew out of two of the four classic 17th-century calculus problems: the tangent line problem and the velocity problem. Both reduce to the same limiting construction.

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Chapter contents

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

10.1

Introduction

Everyone has an intuitive sense of speed as "distance covered per unit time," but that intuition is really about average speed.

10.2

The concept of derivative

Calculus, as a discipline, grew out of four major problems seventeenth-century mathematicians were wrestling with:

10.2.1

The tangent line problem

For a circle, "the tangent at " has a clean, purely geometric meaning: it is the line through perpendicular to the radius at (Fig. 10.1).

10.2.2

Velocity of Rectilinear motion

The same secant-to-tangent limiting idea solves the velocity problem. Suppose an object moves along a straight line with position (signed distance from the origin) at time ; is called the position fun…

10.2.3

The derivative of a Function

The limit that produced both the tangent slope and the instantaneous velocity is important enough to deserve its own name and its own general definition, independent of any particular geometric or phy…

10.2.4

One sided derivatives (left hand and right hand derivatives)

Definition 10.2's limit is two-sided — from both directions at once must give the same value. Splitting it into its two one-sided pieces is what lets us pin down exactly where and why a derivative can…

10.3

Differentiability and Continuity

The definitions above raise an obvious question: can a function be continuous at a point yet still fail to have a derivative there? The book answers this with a sequence of illustrations before provin…

+Exercise 10.1i7 questions
  1. Q1Find the derivatives of the following functions using first principle. (i) $f(x) = 6$ (ii) $f(x) = -4x + 7$ (iii) $f(x) = -x^2 + 2$Free
  2. Q2Find the derivatives from the left and from the right at $x = 1$ (if they exist) of the following functions. Are the functions differentiabl…Free
  3. Q3Determine whether the following function is differentiable at the indicated values. (i) $f(x) = x\,|x|$ at $x = 0$ (ii) $f(x) = |x^2 - 1|$ a…Free
  4. Q4Show that the following functions are not differentiable at the indicated value of $x$. (i) $f(x) = \begin{cases} -x+2, & x \le 2 \\ 2x-4, &…Preview
  5. Q5The graph of $f$ is shown below (Fig. 10.24). State with reasons the $x$ values (the numbers) at which $f$ is not differentiable. *Graph des…Preview
  6. Q6If $f(x) = |x+100| + x^2$, test whether $f'(-100)$ exists.Preview
  7. Q7Examine the differentiability of the following functions in $\mathbb{R}$ by drawing the diagrams. (i) $|\sin x|$ (ii) $|\cos x|$Preview
10.4

Differentiation Rules

Computing every derivative directly from Definition 10.2's limit — "from first principle" — is correct but, as the book notes, "extremely laborious and difficult...

10.4.1

Derivatives of basic elementary functions

With the general rules of §10.4 in hand, we now derive — once, from first principle, each time using the -form — the derivatives of every basic elementary function.

+Exercise 10.2i20 questions
  1. Q1Find the derivative of the following function with respect to the corresponding independent variable: $f(x) = x - 3\sin x$Free
  2. Q2Find the derivative of the following function with respect to the corresponding independent variable: $y = \sin x + \cos x$Free
  3. Q3Find the derivative of the following function with respect to the corresponding independent variable: $f(x) = x \sin x$Free
  4. Q4Find the derivative of the following function with respect to the corresponding independent variable: $y = \cos x - 2\tan x$Preview
  5. Q5Find the derivative of the following function with respect to the corresponding independent variable: $g(t) = t^3\cos t$Preview
  6. Q6Find the derivative of the following function with respect to the corresponding independent variable: $g(t) = 4\sec t + \tan t$Preview
  7. Q7Find the derivative of the following function with respect to the corresponding independent variable: $y = e^x \sin x$Preview
  8. Q8Find the derivative of the following function with respect to the corresponding independent variable: $y = \dfrac{\tan x}{x}$Preview
  9. Q9Find the derivative of the following function with respect to the corresponding independent variable: $y = \dfrac{\sin x}{1+\cos x}$Preview
  10. Q10Find the derivative of the following function with respect to the corresponding independent variable: $y = \dfrac{x}{\sin x + \cos x}$Preview
  11. Q11Find the derivative of the following function with respect to the corresponding independent variable: $y = \dfrac{\tan x - 1}{\sec x}$Preview
  12. Q12Find the derivative of the following function with respect to the corresponding independent variable: $y = \dfrac{\sin x}{x^2}$Preview
  13. Q13Find the derivative of the following function with respect to the corresponding independent variable: $y = \tan\theta(\sin\theta + \cos\thet…Preview
  14. Q14Find the derivative of the following function with respect to the corresponding independent variable: $y = \operatorname{cosec} x \cdot \cot…Preview
  15. Q15Find the derivative of the following function with respect to the corresponding independent variable: $y = x \sin x \cos x$Preview
  16. Q16Find the derivative of the following function with respect to the corresponding independent variable: $y = e^{-x} \cdot \log x$Preview
  17. Q17Find the derivative of the following function with respect to the corresponding independent variable: $y = (x^2+5)\log(1+x)\,e^{-3x}$Preview
  18. Q18Find the derivative of the following function with respect to the corresponding independent variable: $y = \sin x^{\circ}$Preview
  19. Q19Find the derivative of the following function with respect to the corresponding independent variable: $y = \log_{10} x$Preview
  20. Q20Draw the graph of $f'(x)$ if $f(x) = 2x^2 - 5x + 3$.Preview
10.4.2

Examples on Chain Rule

This section is pure practice at recognising the outer function / inner function split that the chain rule (Theorem 10.5) needs, and at combining it with the other rules when several layers or several…

+Exercise 10.3i30 questions
  1. Q1Differentiate the following: $y = (x^2+4x+6)^5$Free
  2. Q2Differentiate the following: $y = \tan 3x$Free
  3. Q3Differentiate the following: $y = \cos(\tan x)$Free
  4. Q4Differentiate the following: $y = \sqrt[3]{1+x^3}$Preview
  5. Q5Differentiate the following: $y = e^{\sqrt{x}}$Preview
  6. Q6Differentiate the following: $y = \sin(e^x)$Preview
  7. Q7Differentiate the following: $F(x) = (x^3+4x)^7$Preview
  8. Q8Differentiate the following: $h(t) = \left(t - \dfrac{1}{t}\right)^{3/2}$Preview
  9. Q9Differentiate the following: $f(t) = \sqrt[3]{1+\tan t}$Preview
  10. Q10Differentiate the following: $y = \cos(a^3+x^3)$Preview
  11. Q11Differentiate the following: $y = e^{-mx}$Preview
  12. Q12Differentiate the following: $y = 4\sec 5x$Preview
  13. Q13Differentiate the following: $y = (2x-5)^4(8x^2-5)^{-3}$Preview
  14. Q14Differentiate the following: $y = (x^2+1)\sqrt[3]{x^2+2}$Preview
  15. Q15Differentiate the following: $y = xe^{-x^2}$Preview
  16. Q16Differentiate the following: $s(t) = \sqrt[4]{\dfrac{t^3+1}{t^3-1}}$Preview
  17. Q17Differentiate the following: $f(x) = \dfrac{x}{\sqrt{7-3x}}$Preview
  18. Q18Differentiate the following: $y = \tan(\cos x)$Preview
  19. Q19Differentiate the following: $y = \dfrac{\sin^2 x}{\cos x}$Preview
  20. Q20Differentiate the following: $y = 5^{-1/x}$Preview
  21. Q21Differentiate the following: $y = \sqrt{1+2\tan x}$Preview
  22. Q22Differentiate the following: $y = \sin^3 x + \cos^3 x$Preview
  23. Q23Differentiate the following: $y = \sin^2(\cos kx)$Preview
  24. Q24Differentiate the following: $y = (1+\cos^2 x)^6$Preview
  25. Q25Differentiate the following: $y = \dfrac{e^{3x}}{1+e^x}$Preview
  26. Q26Differentiate the following: $y = \sqrt{x+\sqrt{x}}$Preview
  27. Q27Differentiate the following: $y = e^{x\cos x}$Preview
  28. Q28Differentiate the following: $y = \sqrt{x+\sqrt{x+\sqrt{x}}}$Preview
  29. Q29Differentiate the following: $y = \sin\!\big(\tan(\sqrt{\sin x})\big)$Preview
  30. Q30Differentiate the following: $y = \sin^{-1}\!\left(\dfrac{1-x^2}{1+x^2}\right)$Preview
10.4.3

Implicit Differentiation

A function in which the dependent variable is expressed solely in terms of the independent variable , i.e. , is called an explicit function — for instance .

10.4.4

Logarithmic Differentiation

Ordinary differentiation rules — the power rule, the exponential rule — each handle only one specific relationship between the base and the exponent (a fixed power of a variable base, or a fixed base…

10.4.5

Substitution method

Some inverse-trigonometric expressions can in principle be differentiated directly by repeated chain-rule and quotient-rule work, but that route is often extremely laborious.

10.4.6

Derivatives of variables defined by parametric equations

So far has been expressed either explicitly in terms of , or implicitly via an equation directly relating and .

10.4.7

Differentiation of one function with respect to another function

The chain rule (Theorem 10.5) differentiates a composite function with respect to the independent variable .

10.4.8

Higher order Derivatives

53 Q

If is the position (displacement) of an object moving along a straight line, its first derivative already has a direct physical meaning: the velocity — this is exactly the instantaneous velocity defin…

+Exercise 10.4i28 questions
  1. Q1Find the derivative of the following: $y = x^{\cos x}$Free
  2. Q2Find the derivative of the following: $y = x^{\log x} + (\log x)^x$Free
  3. Q3Find the derivative of the following: $\sqrt{xy} = e^{(x-y)}$Free
  4. Q4Find the derivative of the following: $x^y = y^x$Preview
  5. Q5Find the derivative of the following: $(\cos x)^{\log x}$Preview
  6. Q6Find the derivative of the following: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$Preview
  7. Q7Find the derivative of the following: $\sqrt{x^2+y^2} = \tan^{-1}\!\left(\dfrac{y}{x}\right)$Preview
  8. Q8Find the derivative of the following: $\tan(x+y)+\tan(x-y) = x$Preview
  9. Q9If $\cos(xy) = x$, show that $\dfrac{dy}{dx} = \dfrac{-(1+y\sin(xy))}{x\sin(xy)}$.Preview
  10. Q10Find the derivative of the following: $\tan^{-1}\sqrt{\dfrac{1-\cos x}{1+\cos x}}$Preview
  11. Q11Find the derivative of the following: $\tan^{-1}\!\left(\dfrac{6x}{1-9x^2}\right)$Preview
  12. Q12Find the derivative of the following: $\cos\!\left(2\tan^{-1}\sqrt{\dfrac{1-x}{1+x}}\right)$Preview
  13. Q13Find the derivative of the following: $x = a\cos^3 t\ ;\ y = a\sin^3 t$Preview
  14. Q14Find the derivative of the following: $x = a(\cos t + t\sin t)\ ;\ y = a(\sin t - t\cos t)$Preview
  15. Q15Find the derivative of the following: $x = \dfrac{1-t^2}{1+t^2},\ y = \dfrac{2t}{1+t^2}$Preview
  16. Q16Find the derivative of the following: $\cos^{-1}\!\left(\dfrac{1-x^2}{1+x^2}\right)$Preview
  17. Q17Find the derivative of the following: $\sin^{-1}(3x-4x^3)$Preview
  18. Q18Find the derivative of the following: $\tan^{-1}\!\left(\dfrac{\cos x + \sin x}{\cos x - \sin x}\right)$Preview
  19. Q19Find the derivative of $\sin x^2$ with respect to $x^2$.Preview
  20. Q20Find the derivative of $\sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right)$ with respect to $\tan^{-1}x$.Preview
  21. Q21If $u = \tan^{-1}\dfrac{\sqrt{1+x^2}-1}{x}$ and $v = \tan^{-1}x$, find $\dfrac{du}{dv}$.Preview
  22. Q22Find the derivative of $\tan^{-1}\!\left(\dfrac{\sin x}{1+\cos x}\right)$ with respect to $\tan^{-1}\!\left(\dfrac{\cos x}{1+\sin x}\right)$…Preview
  23. Q23If $y = \sin^{-1}x$ then find $y''$.Preview
  24. Q24If $y = e^{\tan^{-1}x}$, show that $(1+x^2)y'' + (2x-1)y' = 0$.Preview
  25. Q25If $y = \dfrac{\sin^{-1}x}{\sqrt{1-x^2}}$, show that $(1-x^2)y_2 - 3xy_1 - y = 0$.Preview
  26. Q26If $x = a(\theta+\sin\theta),\ y = a(1-\cos\theta)$ then prove that at $\theta = \dfrac{\pi}{2}$, $y'' = \dfrac{1}{a}$.Preview
  27. Q27If $\sin y = x\sin(a+y)$, then prove that $\dfrac{dy}{dx} = \dfrac{\sin^2(a+y)}{\sin a}$, $a\ne n\pi$.Preview
  28. Q28If $y = (\cos^{-1}x)^2$, prove that $(1-x^2)\dfrac{d^2y}{dx^2} - x\dfrac{dy}{dx} - 2 = 0$. Hence find $y_2$ when $x=0$.Preview
+Exercise 10.5i25 questions
  1. Q1$\dfrac{d}{dx}\left(\dfrac{2}{\pi}\sin x^{\circ}\right)$ is (1) $\dfrac{\pi}{180}\cos x^{\circ}$ (2) $\dfrac{1}{90}\cos x^{\circ}$ (3) $\dfr…Free
  2. Q2If $y = f(x^2+2)$ and $f'(3) = 5$, then $\dfrac{dy}{dx}$ at $x=1$ is (1) 5 (2) 25 (3) 15 (4) 10Free
  3. Q3If $y = \dfrac14 u^4$, $u = \dfrac23 x^3+5$, then $\dfrac{dy}{dx}$ is (1) $\dfrac{1}{27}x^2(2x^3+15)^3$ (2) $\dfrac{2}{27}x(2x^3+5)^3$ (3) $…Free
  4. Q4If $f(x) = x^2-3x$, then the points at which $f(x) = f'(x)$ are (1) both positive integers (2) both negative integers (3) both irrational (4…Preview
  5. Q5If $y = \dfrac{1}{a-z}$, then $\dfrac{dz}{dy}$ is (1) $(a-z)^2$ (2) $-(z-a)^2$ (3) $(z+a)^2$ (4) $-(z+a)^2$Preview
  6. Q6If $y = \cos(\sin x^2)$, then $\dfrac{dy}{dx}$ at $x = \sqrt{\dfrac{\pi}{2}}$ is (1) $-2$ (2) $2$ (3) $-2\sqrt{\dfrac{\pi}{2}}$ (4) $0$Preview
  7. Q7If $y = mx+c$ and $f(0) = f'(0) = 1$, then $f(2)$ is (1) 1 (2) 2 (3) 3 (4) $-3$Preview
  8. Q8If $f(x) = x\tan^{-1}x$, then $f'(1)$ is (1) $1+\dfrac{\pi}{4}$ (2) $\dfrac12+\dfrac{\pi}{4}$ (3) $\dfrac12-\dfrac{\pi}{4}$ (4) $2$Preview
  9. Q9$\dfrac{d}{dx}\left(e^{x+5\log x}\right)$ is (1) $e^x\cdot x^4(x+5)$ (2) $e^x\cdot x(x+5)$ (3) $e^x+\dfrac{5}{x}$ (4) $e^x-\dfrac{5}{x}$Preview
  10. Q10If the derivative of $(ax-5)e^{3x}$ at $x=0$ is $-13$, then the value of $a$ is (1) 8 (2) $-2$ (3) 5 (4) 2Preview
  11. Q11If $x = \dfrac{1-t^2}{1+t^2},\ y = \dfrac{2t}{1+t^2}$ then $\dfrac{dy}{dx}$ is (1) $-\dfrac{y}{x}$ (2) $\dfrac{y}{x}$ (3) $-\dfrac{x}{y}$ (4…Preview
  12. Q12If $x = a\sin\theta$ and $y = b\cos\theta$, then $\dfrac{d^2y}{dx^2}$ is (1) $\dfrac{a}{b^2}\sec^2\theta$ (2) $-\dfrac{b}{a}\sec^2\theta$ (3…Preview
  13. Q13The differential coefficient of $\log_{10}x$ with respect to $\log_x 10$ is (1) 1 (2) $-(\log_{10}x)^2$ (3) $(\log_x 10)^2$ (4) $\dfrac{x^2}…Preview
  14. Q14If $f(x) = x+2$, then $f'(f(x))$ at $x=4$ is (1) 8 (2) 1 (3) 4 (4) 5Preview
  15. Q15If $y = \dfrac{(1-x)^2}{x^2}$, then $\dfrac{dy}{dx}$ is (1) $\dfrac{2}{x^2}+\dfrac{2}{x^3}$ (2) $-\dfrac{2}{x^2}+\dfrac{2}{x^3}$ (3) $-\dfra…Preview
  16. Q16If $pv = 81$, then $\dfrac{dp}{dv}$ at $v=9$ is (1) 1 (2) $-1$ (3) 2 (4) $-2$Preview
  17. Q17If $f(x) = \begin{cases} x-5, & x\le 1 \\ 4x^2-9, & 1<x<2 \\ 3x+4, & x\ge 2 \end{cases}$, then the right hand derivative of $f(x)$ at $x=2$…Preview
  18. Q18It is given that $f'(a)$ exists, then $\displaystyle\lim_{x\to a}\dfrac{xf(a)-af(x)}{x-a}$ is (1) $f(a)-af'(a)$ (2) $f'(a)$ (3) $-f'(a)$ (4)…Preview
  19. Q19If $f(x) = \begin{cases} x+1, & x<2 \\ 2x-1, & x\ge 2 \end{cases}$, then $f'(2)$ is (1) 0 (2) 1 (3) 2 (4) does not existPreview
  20. Q20If $g(x) = (x^2+2x+3)f(x)$ and $f(0) = 5$ and $\displaystyle\lim_{x\to0}\dfrac{f(x)-5}{x}=4$, then $g'(0)$ is (1) 20 (2) 14 (3) 18 (4) 12Preview
  21. Q21If $f(x) = \begin{cases} x+2, & -1<x<3 \\ 5, & x=3 \\ 8-x, & x>3 \end{cases}$, then at $x=3$, $f'(x)$ is (1) 1 (2) $-1$ (3) 0 (4) does not e…Preview
  22. Q22The derivative of $f(x) = x|x|$ at $x=-3$ is (1) 6 (2) $-6$ (3) does not exist (4) 0Preview
  23. Q23If $f(x) = \begin{cases} 2a-x, & -a<x<a \\ 3x-2a, & x\ge a \end{cases}$, then which one of the following is true? (1) $f(x)$ is not differen…Preview
  24. Q24If $f(x) = \begin{cases} ax^2-b, & -1<x<1 \\ \dfrac{1}{|x|}, & \text{elsewhere} \end{cases}$ is differentiable at $x=1$, then (1) $a=\dfrac1…Preview
  25. Q25The number of points in $\mathbb{R}$ in which the function $f(x) = |x-1|+|x-3|+\sin x$ is not differentiable, is (1) 3 (2) 2 (3) 1 (4) 4Preview

Sample & Board Papers

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

+Show 33 questions33 questions
  1. Q1Which of the function is not differentiable? (a) $f(x) = \sin x + \cos x$ in $(-\infty, \infty)$ (b) $f(x) = \sin x$ in $(-\infty, \infty)$…Preview
  2. Q2Find $f'(x)$, if $f(x) = \sin|x|$, by removing the modulus sign.Preview
  3. Q3Find $\dfrac{dy}{dx}$ if $\tan(x+y) + \tan(x-y) = 1$.Preview
  4. Q4(a) Draw the graph of the function $f(x) = \begin{cases} 2x, & x<1 \\ 2, & x=1 \\ x+1, & x>1 \end{cases}$ and state the differentiability at…Preview
  5. Q5If $f(x)=\begin{cases}2a-x, & -a<x<a\\ 3x-2a, & x\ge a\end{cases}$ then which one of the following is true? (a) $f(x)$ is continuous for all…Preview
  6. Q6If $f(x)=x^2-3x$, then the points at which $f(x)=f'(x)$ are: (a) both irrational (b) one rational and another irrational (c) both positive i…Preview
  7. Q7Differentiate: $y=\sin^{-1}\left(\dfrac{1-x^2}{1+x^2}\right)$Preview
  8. Q8Find $\dfrac{dy}{dx}$ if $x=a(t-\sin t)$, $y=a(1-\cos t)$.Preview
  9. Q9(a) Find $\dfrac{d^2y}{dx^2}$ if $x^2+y^2=4$. **OR** (b) The chances of X, Y and Z becoming managers of a certain company are 4 : 2 : 3. The…Preview
  10. Q10The number of points in $\mathbf{R}$ in which the function $f(x) = |x-1| + |x-3| + \sin x$ is not differentiable, is: (a) 3 (b) 2 (c) 1 (d)…Preview
  11. Q11If $f(x) = x\tan^{-1}x$ then, $f'(0) + f'(1)$ is: (a) $1 + \dfrac{\pi}{4}$ (b) $\dfrac{1}{2} + \dfrac{\pi}{4}$ (c) $\dfrac{1}{2} - \dfrac{\p…Preview
  12. Q12Differentiate $y = \dfrac{x}{1+\tan x}$ with respect to '$x$'.Preview
  13. Q13If $y = \tan^{-1}\left(\dfrac{1-x^2}{1+x^2}\right)$ find $y'$.Preview
  14. Q14If $y=f(x^2+2)$ and $f'(3)=5$ then, $\dfrac{dy}{dx}$ at $x=1$ is: (a) 15 (b) 5 (c) 10 (d) 25Preview
  15. Q15If $f(x) = \begin{cases} x+2, & -1<x<3 \\ 5, & x=3 \\ 8-x, & x>3 \end{cases}$, then at $x=3$, $f'(x)$ is: (a) 0 (b) 1 (c) does not exist (d)…Preview
  16. Q16Differentiate $y=x^3+5x^2+3x+7$ with respect to $x$.Preview
  17. Q17Find $\dfrac{dy}{dx}$ if $x^2+y^2=1$.Preview
  18. Q18If $f(x) = mx + c$ and $f(0) = f'(0) = 1$ then $f(3)$ is: (a) 3 (b) 1 (c) 4 (d) 2Preview
  19. Q19Find $\sqrt[3]{1001}$ approximately (two decimal places).Preview
  20. Q20Differentiate with respect to $x$. $y = \dfrac{\cos x}{x^3}$Preview
  21. Q21If $y=f(x^2+2)$ and $f'(3)=5$, then $\dfrac{dy}{dx}$ at $x=1$ is: (a) $15$ (b) $5$ (c) $10$ (d) $25$Preview
  22. Q22If $y=e^{\sin x}$ then $\dfrac{dy}{dx}=$ (a) $\sin x\,e^{\sin x}$ (b) $e^{\sin x}$ (c) $\cos x\,e^{\sin x}$ (d) $e^{\cos x}$Preview
  23. Q23Find $f''$ if $f(x)=x\cos x$.Preview
  24. Q24Differentiate the following with respect to $x$. $y=xe^x\log x$Preview
  25. Q25The derivative of $f(x) = x|x|$ at $x = -3$ is: (a) does not exist (b) $6$ (c) $0$ (d) $-6$Preview
  26. Q26$\dfrac{d}{dx}\left(\dfrac{2}{\pi}\sin x^\circ\right)$ is: (a) $\dfrac{\pi}{90}\cos x^\circ$ (b) $\dfrac{\pi}{180}\cos x^\circ$ (c) $\dfrac{…Preview
  27. Q27Differentiate : $y = e^{\sin x}$Preview
  28. Q28Find $\dfrac{dy}{dx}$, if $y = \cos^{-1}(2\cos^2 x - 1)$Preview
  29. Q29If $y = mx + c$ and $f(0) = f'(0) = 1$, then $f(2)$ is: (a) 3 (b) 1 (c) -3 (d) 2Preview
  30. Q30Find $f'(7)$ if $f(x) = |x - 5|$ (a) -1 (b) 1 (c) 5 (d) 7Preview
  31. Q31Find the value of $\sqrt[3]{65}$Preview
  32. Q32Find $\dfrac{dy}{dx}$ if $x=a(t-\sin t)$, $y=a(1-\cos t)$Preview
  33. Q33If $y=e^{\tan^{-1}x}$, show that $(1+x^2)y''+(2x-1)y'=0$ **OR** Show that $\begin{vmatrix}b+c & a-c & a-b\\ b-c & c+a & b-a\\ c-b & c-a & a+…Preview