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Mathematics · Class 12 Science

Ch 5Continuity and Differentiability — Class 12 Mathematics, concept-first.

This chapter builds directly on the differentiation you studied in Class XI, where you learned to find derivatives of polynomial and trigonometric functions. Now we take a deeper look at the ideas that make differentiation possible.

281

Q&A

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Continuity At A Point

Imagine drawing the graph of a function and putting your pen down at . If the function is continuous there, you can draw straight through that point without lifting your pen — no jump, no hole, no break.

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5.1

Introduction

This chapter builds directly on the differentiation you studied in Class XI, where you learned to find derivatives of polynomial and trigonometric functions.

5.2

Continuity

Before we formalise the idea, consider two simple functions that illustrate what it means for a function to fail to be continuous at a point.

+Worked Examplesi15 questions
  1. Example 1Check the continuity of the function $f$ given by $f(x) = 2x + 3$ at $x = 1$.Free
  2. Example 2Examine whether the function $f$ given by $f(x) = x^2$ is continuous at $x = 0$.Free
  3. Example 3Discuss the continuity of the function $f$ given by $f(x) = |x|$ at $x = 0$.Free
  4. Example 4Show that the function $f$ given by $f(x) = \begin{cases} x^3 + 3, & \text{if } x \neq 0 \\ 1, & \text{if } x = 0 \end{cases}$ is not contin…Preview
  5. Example 5Check the points where the constant function $f(x) = k$ is continuous.Preview
  6. Example 6Prove that the identity function on real numbers given by $f(x) = x$ is continuous at every real number.Preview
  7. Example 7Is the function defined by $f(x) = |x|$, a continuous function?Preview
  8. Example 8Discuss the continuity of the function $f$ given by $f(x) = x^3 + x^2 - 1$.Preview
  9. Example 9Discuss the continuity of the function $f$ defined by $f(x) = \frac{1}{x}$, $x \neq 0$.Preview
  10. Example 10Discuss the continuity of the function $f$ defined by $f(x) = \begin{cases} x + 2, & \text{if } x \leq 1 \\ x - 2, & \text{if } x > 1 \end{c…Preview
  11. Example 11Find all the points of discontinuity of the function $f$ defined by $f(x) = \begin{cases} x + 2, & \text{if } x < 1 \\ 0, & \text{if } x = 1…Preview
  12. Example 12Discuss the continuity of the function defined by $f(x) = \begin{cases} x + 2, & \text{if } x < 0 \\ -x + 2, & \text{if } x > 0 \end{cases}$…Preview
  13. Example 13Discuss the continuity of the function $f$ given by $f(x) = \begin{cases} x, & \text{if } x \geq 0 \\ x^2, & \text{if } x < 0 \end{cases}$.Preview
  14. Example 14Show that every polynomial function is continuous.Preview
  15. Example 15Find all the points of discontinuity of the greatest integer function defined by $f(x) = [x]$, where $[x]$ denotes the greatest integer less…Preview
5.2.1

Algebra of Continuous Functions

39 Q

Since continuity at a point is defined entirely by the limit at that point, continuous functions inherit the algebra of limits.

+Worked Examplesi5 questions
  1. Example 16A rational function is a function of the form $f(x) = \dfrac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomial functions of $x$ and $q(x)…Free
  2. Example 17Discuss the continuity of sine function.Free
  3. Example 18Prove that the function defined by $f(x) = \tan x$ is a continuous function.Preview
  4. Example 19Show that the function defined by $f(x) = \sin(x^2)$ is a continuous function.Preview
  5. Example 20Show that the function $f$ defined by $f(x) = |1 - x + |x||$, where $x$ is any real number, is a continuous function.Preview
+Exercise 5.1i34 questions
  1. Q1Prove that the function $f(x) = 5x - 3$ is continuous at $x = 0$, at $x = -3$ and at $x = 5$.Free
  2. Q2Examine the continuity of the function $f(x) = 2x^2 - 1$ at $x = 3$.Free
  3. Q3Examine the following functions for continuity. (a) $f(x) = x - 5$ (b) $f(x) = \frac{1}{x-5}$, $x \neq 5$ (c) $f(x) = \frac{x^2 - 25}{x+5}$,…Free
  4. Q4Prove that the function $f(x) = x^n$ is continuous at $x = n$, where $n$ is a positive integer.Preview
  5. Q5Is the function $f$ defined by $f(x) = \begin{cases} x, & \text{if } x \leq 1 \\ 5, & \text{if } x > 1 \end{cases}$ continuous at $x = 0$? A…Preview
  6. Q6Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} 2x+3, & \text{if } x \leq 2 \\ 2x-3, & \text{if } x >…Preview
  7. Q7Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} |x|+3, & \text{if } x \leq -3 \\ -2x, & \text{if } -3…Preview
  8. Q8Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} \frac{|x|}{x}, & \text{if } x \neq 0 \\ 0, & \text{if…Preview
  9. Q9Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} \frac{x}{|x|}, & \text{if } x < 0 \\ -1, & \text{if }…Preview
  10. Q10Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} x+1, & \text{if } x \geq 1 \\ x^2+1, & \text{if } x <…Preview
  11. Q11Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} x^3-3, & \text{if } x \leq 2 \\ x^2+1, & \text{if } x…Preview
  12. Q12Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} x^{10}-1, & \text{if } x \leq 1 \\ x^2, & \text{if }…Preview
  13. Q13Is the function defined by $f(x) = \begin{cases} x+5, & \text{if } x \leq 1 \\ x-5, & \text{if } x > 1 \end{cases}$ a continuous function?Preview
  14. Q14Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \begin{cases} 3, & \text{if } 0 \le x \le 1 \\ 4, & \text{if } 1…Preview
  15. Q15Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \begin{cases} 2x, & \text{if } x < 0 \\ 0, & \text{if } 0 \le x…Preview
  16. Q16Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \begin{cases} -2, & \text{if } x \le -1 \\ 2x, & \text{if } -1 <…Preview
  17. Q17Find the relationship between $a$ and $b$ so that the function $f$ defined by $f(x) = \begin{cases} ax+1, & \text{if } x \le 3 \\ bx+3, & \t…Preview
  18. Q18For what value of $\lambda$ is the function defined by $f(x) = \begin{cases} \lambda(x^2-2x), & \text{if } x \le 0 \\ 4x+1, & \text{if } x >…Preview
  19. Q19Show that the function defined by $g(x) = x - [x]$ is discontinuous at all integral points. Here $[x]$ denotes the greatest integer less tha…Preview
  20. Q20Is the function defined by $f(x) = x^2 - \sin x + 5$ continuous at $x = \pi$?Preview
  21. Q21Discuss the continuity of the following functions: (a) $f(x) = \sin x + \cos x$ (b) $f(x) = \sin x - \cos x$ (c) $f(x) = \sin x \cdot \cos x…Preview
  22. Q22Discuss the continuity of the cosine, cosecant, secant and cotangent functions.Preview
  23. Q23Find all points of discontinuity of $f$, where $f(x) = \begin{cases} \frac{\sin x}{x}, & \text{if } x < 0 \\ x+1, & \text{if } x \ge 0 \end{…Preview
  24. Q24Determine if $f$ defined by $f(x) = \begin{cases} x^2 \sin \frac{1}{x}, & \text{if } x \ne 0 \\ 0, & \text{if } x = 0 \end{cases}$ is a cont…Preview
  25. Q25Examine the continuity of $f$, where $f$ is defined by $f(x) = \begin{cases} \sin x - \cos x, & \text{if } x \neq 0 \\ -1, & \text{if } x =…Preview
  26. Q26Find the values of $k$ so that the function $f$ is continuous at the indicated point, where $f$ is defined by $f(x) = \begin{cases} \frac{k…Preview
  27. Q27Find the values of $k$ so that the function $f$ is continuous at the indicated point, where $f$ is defined by $f(x) = \begin{cases} kx^2, &…Preview
  28. Q28Find the values of $k$ so that the function $f$ is continuous at the indicated point, where $f$ is defined by $f(x) = \begin{cases} kx+1, &…Preview
  29. Q29Find the values of $k$ so that the function $f$ is continuous at the indicated point, where $f$ is defined by $f(x) = \begin{cases} kx+1, &…Preview
  30. Q30Find the values of $a$ and $b$ such that the function defined by $f(x) = \begin{cases} 5, & \text{if } x \leq 2 \\ ax+b, & \text{if } 2 < x…Preview
  31. Q31Show that the function defined by $f(x) = \cos (x^2)$ is a continuous function.Preview
  32. Q32Show that the function defined by $f(x) = |\cos x|$ is a continuous function.Preview
  33. Q33Examine that $\sin |x|$ is a continuous function.Preview
  34. Q34Find all the points of discontinuity of $f$ defined by $f(x) = |x| - |x+1|$.Preview
5.3

Differentiability

The idea of differentiability builds directly on the derivative. Before we explore when a function fails to be differentiable, recall exactly what it means for a derivative to exist.

5.3.1

Derivatives of Composite Functions

11 Q

For a function like you could expand and differentiate term by term, but that is hopeless for . The better view is that is two functions nested: with and , we have , a composite function ( applied fir…

+Worked Examplesi1 question
  1. Example 21Find the derivative of the function given by $f(x) = \sin(x^2)$.Preview
+Exercise 5.2i10 questions
  1. Q1Find $\frac{dy}{dx}$ in the following: $\sin (x^2 + 5)$Free
  2. Q2Find $\frac{dy}{dx}$ in the following: $\cos (\sin x)$Free
  3. Q3Find $\frac{dy}{dx}$ in the following: $\sin (ax + b)$Free
  4. Q4Find $\frac{dy}{dx}$ in the following: $\sec (\tan (\sqrt{x}))$Preview
  5. Q5Find $\frac{dy}{dx}$ in the following: $\frac{\sin (ax + b)}{\cos (cx + d)}$Preview
  6. Q6Find $\frac{dy}{dx}$ in the following: $\cos x^3 \cdot \sin^2 (x^5)$Preview
  7. Q7Find $\frac{dy}{dx}$ in the following: $2\sqrt{\cot(x^2)}$Preview
  8. Q8Find $\frac{dy}{dx}$ in the following: $\cos(\sqrt{x})$Preview
  9. Q9Prove that the function $f$ given by $f(x) = |x - 1|, x \in \mathbf{R}$ is not differentiable at $x = 1$.Preview
  10. Q10Prove that the greatest integer function defined by $f(x) = [x]$, $0 < x < 3$, is not differentiable at $x = 1$ and $x = 2$.Preview
5.3.2

Derivatives of Implicit Functions

So far you have differentiated functions in the form — an explicit function, where is given directly in terms of (e.g. or ). But many relations are not in this solved form. Consider:

+Worked Examplesi2 questions
  1. Example 22Find $\frac{dy}{dx}$ if $x - y = \pi$.Free
  2. Example 23Find $\frac{dy}{dx}$, if $y + \sin y = \cos x$.Preview
5.3.3

Derivatives of Inverse Trigonometric Functions

16 Q

Inverse trigonometric functions are continuous on their domains (accepted without proof). To differentiate them we combine the chain rule with implicit differentiation: if then , and differentiating b…

+Worked Examplesi1 question
  1. Example 24Find the derivative of $f$ given by $f(x) = \sin^{-1} x$ assuming it exists.Preview
+Exercise 5.3i15 questions
  1. Q1Find $\frac{dy}{dx}$ in the following: $2x + 3y = \sin x$Free
  2. Q2Find $\frac{dy}{dx}$ in the following: $2x + 3y = \sin y$Free
  3. Q3Find $\frac{dy}{dx}$ in the following: $ax + by^2 = \cos y$Free
  4. Q4Find $\frac{dy}{dx}$ in the following: $xy + y^2 = \tan x + y$Preview
  5. Q5Find $\frac{dy}{dx}$ in the following: $x^2 + xy + y^2 = 100$Preview
  6. Q6Find $\frac{dy}{dx}$ in the following: $x^3 + x^2y + xy^2 + y^3 = 81$Preview
  7. Q7Find $\frac{dy}{dx}$ in the following: $\sin^2 y + \cos xy = \kappa$Preview
  8. Q8Find $\frac{dy}{dx}$ in the following: $\sin^2 x + \cos^2 y = 1$Preview
  9. Q9Find $\frac{dy}{dx}$ in the following: $y = \sin^{-1} \left(\frac{2x}{1+x^2}\right)$Preview
  10. Q10Find $\frac{dy}{dx}$ in the following: $y = \tan^{-1} \left(\frac{3x-x^3}{1-3x^2}\right), -\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}$Preview
  11. Q11Find $\frac{dy}{dx}$ in the following: $y = \cos^{-1} \left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$Preview
  12. Q12Find $\frac{dy}{dx}$ in the following: $y = \sin^{-1} \left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$Preview
  13. Q13Find $\frac{dy}{dx}$ in the following: $y = \cos^{-1} \left(\frac{2x}{1+x^2}\right), -1 < x < 1$Preview
  14. Q14Find $\frac{dy}{dx}$ in the following: $y = \sin^{-1} \left(2x\sqrt{1-x^2}\right), -\frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}$Preview
  15. Q15Find $\frac{dy}{dx}$ in the following: $y = \sec^{-1} \left(\frac{1}{2x^2-1}\right), 0 < x < \frac{1}{\sqrt{2}}$Preview
5.4

Exponential and Logarithmic Functions

12 Q

Polynomial functions grow at a rate set by their degree: for , rises faster as increases. This raises a question: is there a function that grows faster than any polynomial, no matter how high the degr…

+Worked Examplesi2 questions
  1. Example 25Is it true that $x = e^{\log x}$ for all real $x$?Free
  2. Example 26Differentiate the following w.r.t. $x$: (i) $e^{-x}$ (ii) $\sin(\log x)$, $x > 0$ (iii) $\cos^{-1}(e^x)$ (iv) $e^{\cos x}$.Preview
+Exercise 5.4i10 questions
  1. Q1Find $\frac{dy}{dx}$ in the following: $ \frac{e^x}{\sin x} $Free
  2. Q2Differentiate the following with respect to $x$: $ e^{\sin^{-1} x} $Free
  3. Q3Differentiate the following w.r.t. $x$: $e^{x^3}$Free
  4. Q4Differentiate the following with respect to $x$: $ \sin (\tan^{-1} e^{-x}) $Preview
  5. Q5Find $\frac{dy}{dx}$ in the following: $ \log (\cos e^x) $Preview
  6. Q6Find $\frac{dy}{dx}$ in the following: $ e^x + e^{x^2} + ... + e^{x^5} $Preview
  7. Q7Find $\frac{dy}{dx}$ in the following: $ \sqrt{e^{\sqrt{x}}}, x > 0 $Preview
  8. Q8Find $\frac{dy}{dx}$ in the following: $ \log (\log x), x > 1 $Preview
  9. Q9Find $\frac{dy}{dx}$ in the following: $ \frac{\cos x}{\log x}, x > 0 $Preview
  10. Q10Differentiate the function $\cos(\log x + e^x)$ with respect to $x$, where $x > 0$.Preview
5.5

Logarithmic Differentiation

22 Q

The standard rules handle (variable base, constant exponent) and (constant base, variable exponent). But for , where both base and exponent are functions of , neither the power rule nor the exponentia…

+Worked Examplesi4 questions
  1. Example 27Differentiate $\sqrt{\dfrac{(x-3)(x^2+4)}{3x^2+4x+5}}$ w.r.t. $x$.Free
  2. Example 28Differentiate $a^x$ w.r.t. $x$, where $a$ is a positive constant.Free
  3. Example 29Differentiate $x^{\sin x}$, $x > 0$ w.r.t. $x$.Preview
  4. Example 30Find $\dfrac{dy}{dx}$, if $y^x + x^y + x^x = a^b$, where $a$ and $b$ are positive constants.Preview
+Exercise 5.5i18 questions
  1. Q1Find $\frac{dy}{dx}$ in the following: $\cos x \cdot \cos 2x \cdot \cos 3x$Free
  2. Q2Find $\frac{dy}{dx}$ in the following: $\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$Free
  3. Q3Find $\frac{dy}{dx}$ in the following: $(\log x)^{\cos x}$Free
  4. Q4Find $\frac{dy}{dx}$ in the following: $x^x - 2^{\sin x}$Preview
  5. Q5Find $\frac{dy}{dx}$ in the following: $(x+3)^2 \cdot (x+4)^3 \cdot (x+5)^4$Preview
  6. Q6Find $\frac{dy}{dx}$ in the following: $\left(x+\frac{1}{x}\right)^x + x^{\left(1+\frac{1}{x}\right)}$Preview
  7. Q7Find $\frac{dy}{dx}$ in the following: $(\log x)^x + x^{\log x}$Preview
  8. Q8Differentiate the function $(\sin x)^x + \sin^{-1}\sqrt{x}$ with respect to $x$.Preview
  9. Q9Find $\frac{dy}{dx}$ in the following: $x^{\sin x} + (\sin x)^{\cos x}$Preview
  10. Q10Find $\frac{dy}{dx}$ in the following: $x^{\cos x} + \frac{x^2+1}{x^2-1}$Preview
  11. Q11Differentiate the function $(x\cos x)^x + (x\sin x)^{\frac{1}{x}}$ with respect to $x$.Preview
  12. Q12Find $\frac{dy}{dx}$ in the following: $x^y + y^x = 1$Preview
  13. Q13Find $\frac{dy}{dx}$ in the following: $y^x = x^y$Preview
  14. Q14Find $\frac{dy}{dx}$ in the following: $(\cos x)^y = (\cos y)^x$Preview
  15. Q15Find $\frac{dy}{dx}$ in the following: $xy = e^{(x-y)}$Preview
  16. Q16Find the derivative of the function given by $f(x) = (1+x)(1+x^2)(1+x^4)(1+x^8)$ and hence find $f'(1)$.Preview
  17. Q17Differentiate $(x^2 - 5x + 8)(x^3 + 7x + 9)$ in three ways mentioned below: (i) by using product rule (ii) by expanding the product to obtai…Preview
  18. Q18If $u, v$ and $w$ are functions of $x$, then show that $\frac{d}{dx} (u \cdot v \cdot w) = \frac{du}{dx} \cdot v \cdot w + u \cdot \frac{dv}…Preview
5.6

Derivatives of Functions in Parametric Forms

15 Q

Sometimes the relationship between and is given neither explicitly as nor implicitly as . Instead, both and are expressed separately in terms of a third variable — the parameter — which links them thr…

+Worked Examplesi4 questions
  1. Example 31Find $\frac{dy}{dx}$, if $x = a\cos\theta$, $y = a\sin\theta$.Free
  2. Example 32Find $\frac{dy}{dx}$, if $x = at^2$, $y = 2at$.Free
  3. Example 33Find $\frac{dy}{dx}$, if $x = a(\theta + \sin\theta)$, $y = a(1 - \cos\theta)$.Preview
  4. Example 34Find $\frac{dy}{dx}$, if $x^{2/3} + y^{2/3} = a^{2/3}$.Preview
+Exercise 5.6i11 questions
  1. Q1Find $\frac{dy}{dx}$ in the following: $x = 2at^2, y = at^4$Free
  2. Q2Find $\frac{dy}{dx}$ in the following: $x = a \cos \theta, y = b \cos \theta$Free
  3. Q3Find $\frac{dy}{dx}$ in the following: $x = \sin t, y = \cos 2t$Free
  4. Q4Find $\frac{dy}{dx}$ in the following: $x = 4t, y = \frac{4}{t}$Preview
  5. Q5Find $\frac{dy}{dx}$ in the following: $x = \cos \theta - \cos 2\theta, y = \sin \theta - \sin 2\theta$Preview
  6. Q6Find $\frac{dy}{dx}$ in the following: $x = a (\theta - \sin \theta), y = a (1 + \cos \theta)$Preview
  7. Q7Find $\frac{dy}{dx}$ in the following: $x = \frac{\sin^3 t}{\sqrt{\cos 2t}}, y = \frac{\cos^3 t}{\sqrt{\cos 2t}}$Preview
  8. Q8Find $\frac{dy}{dx}$ in the following: $x = a \left(\cos t + \log \tan \frac{t}{2}\right), y = a \sin t$Preview
  9. Q9Find $\frac{dy}{dx}$ in the following: $x = a \sec \theta, y = b \tan \theta$Preview
  10. Q10Find $\frac{dy}{dx}$ in the following: $x = a (\cos \theta + \theta \sin \theta), y = a (\sin \theta - \theta \cos \theta)$Preview
  11. Q11If $x = \sqrt{a^{\sin^{-1} t}}, y = \sqrt{a^{\cos^{-1} t}}$, show that $\frac{dy}{dx} = -\frac{y}{x}$Preview
5.7

Second Order Derivative

21 Q

For , the first derivative tells us how fast changes as changes. But is itself just another function of — so we can ask the same question about it: how fast is changing? Differentiating once more with…

+Worked Examplesi4 questions
  1. Example 35Find $\frac{d^2y}{dx^2}$, if $y = x^3 + \tan x$.Free
  2. Example 36If $y = A\sin x + B\cos x$, then prove that $\frac{d^2y}{dx^2} + y = 0$.Free
  3. Example 37If $y = 3e^{2x} + 2e^{3x}$, prove that $\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = 0$.Preview
  4. Example 38If $y = \sin^{-1} x$, show that $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = 0$.Preview
+Exercise 5.7i17 questions
  1. Q1Find the second order derivative of the function $x^2 + 3x + 2$.Free
  2. Q2Find the second order derivative of the function: $x^{20}$Free
  3. Q3Find $\frac{dy}{dx}$ in the following: $ x \cdot \cos x $Free
  4. Q4Find the second order derivative of the function: $\log x$Preview
  5. Q5Find $\frac{dy}{dx}$ in the following: $ x^3 \log x $Preview
  6. Q6Find $\frac{dy}{dx}$ in the following: $ e^x \sin 5x $Preview
  7. Q7Find $\frac{dy}{dx}$ in the following: $ e^{6x} \cos 3x $Preview
  8. Q8Find the second order derivative of the function $ \tan^{-1} x $.Preview
  9. Q9Find $\frac{dy}{dx}$ in the following: $ \log (\log x) $Preview
  10. Q10Find $\frac{dy}{dx}$ in the following: $ \sin (\log x) $Preview
  11. Q11If $y = 5 \cos x - 3 \sin x$, prove that $ \frac{d^2 y}{dx^2} + y = 0 $Preview
  12. Q12If $y = \cos^{-1} x$, Find $\frac{d^2 y}{dx^2}$ in terms of $y$ alone.Preview
  13. Q13If $y = 3 \cos (\log x) + 4 \sin (\log x)$, show that $x^2 y_2 + xy_1 + y = 0$.Preview
  14. Q14If $y = Ae^{mx} + Be^{nx}$, show that $\frac{d^2 y}{dx^2} - (m+n)\frac{dy}{dx} + mny = 0$.Preview
  15. Q15If $y = 500e^{7x} + 600e^{-7x}$, show that $\frac{d^2 y}{dx^2} = 49y$.Preview
  16. Q16If $e^y (x+1) = 1$, show that $\frac{d^2 y}{dx^2} = \left(\frac{dy}{dx}\right)^2$.Preview
  17. Q17If $y = (\tan^{-1} x)^2$, show that $(x^2+1)^2 y_2 + 2x(x^2+1) y_1 = 2$. Miscellaneous ExamplesPreview

Miscellaneous Examples

+Miscellaneous Examplesi5 questions
  1. Example 39Differentiate w.r.t. $x$, the following function: (i) $\sqrt{3x+2} + \dfrac{1}{\sqrt{2x^2+4}}$ (ii) $\log_7(\log x)$.Free
  2. Example 40Differentiate the following w.r.t. $x$: (i) $\cos^{-1}(\sin x)$ (ii) $\tan^{-1}\left(\dfrac{\sin x}{1 + \cos x}\right)$ (iii) $\sin^{-1}\lef…Free
  3. Example 41Find $f'(x)$ if $f(x) = (\sin x)^{\sin x}$ for all $0 < x < \pi$.Preview
  4. Example 42For a positive constant $a$, find $\frac{dy}{dx}$, where $y = a^{t + \frac{1}{t}}$, and $x = \left(t + \frac{1}{t}\right)^a$.Preview
  5. Example 43Differentiate $\sin^2 x$ w.r.t. $e^{\cos x}$.Preview

Miscellaneous Exercise on Chapter 5

+Miscellaneous Exercisei22 questions
  1. Q1Find $\frac{dy}{dx}$ in the following: $(3x^2 - 9x + 5)^9$Free
  2. Q2Differentiate the function $\sin^3 x + \cos^6 x$ with respect to $x$.Free
  3. Q3Find $\frac{dy}{dx}$ in the following: $(5x)^{3 \cos 2x}$Free
  4. Q4Differentiate the function $\sin^{-1}(x \sqrt{x}), 0 \leq x \leq 1$ with respect to $x$.Preview
  5. Q5Differentiate the function $\dfrac{\cos^{-1}\left(\frac{x}{2}\right)}{\sqrt{2x+7}}$, $-2 < x < 2$, with respect to $x$.Preview
  6. Q6Differentiate the function $\cot^{-1} \left[\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\right], 0 < x < \fra…Preview
  7. Q7Find $\frac{dy}{dx}$ in the following: $(\log x)^{\log x}, x > 1$Preview
  8. Q8Find $\frac{dy}{dx}$ in the following: $\cos (a \cos x + b \sin x)$, for some constant $a$ and $b$.Preview
  9. Q9Find $\frac{dy}{dx}$ in the following: $(\sin x - \cos x)^{(\sin x - \cos x)}, \frac{\pi}{4} < x < \frac{3\pi}{4}$Preview
  10. Q10Find $\frac{dy}{dx}$ in the following: $x^x + x^a + a^x + a^a$, for some fixed $a > 0$ and $x > 0$Preview
  11. Q11Find $\frac{dy}{dx}$ in the following: $x^{x-3} + (x-3)^x$, for $x > 3$Preview
  12. Q12Find $\frac{dy}{dx}$, if $y = 12 (1 - \cos t)$, $x = 10 (t - \sin t)$, $-\frac{\pi}{2} < t < \frac{\pi}{2}$Preview
  13. Q13Find $\frac{dy}{dx}$, if $y = \sin^{-1} x + \sin^{-1} \sqrt{1-x^2}$, $0 < x < 1$Preview
  14. Q14If $x \sqrt{1+y} + y \sqrt{1+x} = 0$, for $-1 < x < 1$, prove that $\frac{dy}{dx} = -\frac{1}{(1+x)^2}$Preview
  15. Q15If $(x-a)^2 + (y-b)^2 = c^2$, for some $c > 0$, prove that $\frac{\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2y}{dx^…Preview
  16. Q16If $\cos y = x \cos (a+y)$, with $\cos a \neq \pm 1$, prove that $\frac{dy}{dx} = \frac{\cos^2 (a+y)}{\sin a}$.Preview
  17. Q17If $x = a (\cos t + t \sin t)$ and $y = a (\sin t - t \cos t)$, find $\frac{d^2y}{dx^2}$.Preview
  18. Q18If $f(x) = |x|^3$, show that $f''(x)$ exists for all real $x$ and find it.Preview
  19. Q19Using the fact that $\sin (A+B) = \sin A \cos B + \cos A \sin B$ and the differentiation, obtain the sum formula for cosines.Preview
  20. Q20Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.Preview
  21. Q21If $y = \begin{vmatrix} f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \end{vmatrix}$, prove that $\frac{dy}{dx} = \begin{vmatrix} f'(x) & g'(…Preview
  22. Q22If $y = e^{a \cos^{-1} x}$, $-1 \leq x \leq 1$, show that $(1-x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} - a^2 y = 0$.Preview

Summary

- Continuity at a point: A function is continuous at if . This requires three conditions: is defined, exists, and both are equal.

NCERT Exemplar

Higher-order thinking problems from the NCERT Exemplar.

+Show 91 questions91 questions
  1. Q1Examine the continuity of the function $f(x) = x^3 + 2x^2 - 1$ at $x = 1$.Free
  2. Q2Find whether the function is continuous or discontinuous at the indicated point: $f(x) = \begin{cases} 3x + 5, & x \ge 2 \\ x^2, & x < 2 \en…Free
  3. Q3Find whether the function is continuous or discontinuous at the indicated point: $f(x) = \begin{cases} \dfrac{1 - \cos 2x}{x^2}, & x \ne 0 \…Free
  4. Q4Find whether the function is continuous or discontinuous at the indicated point: $f(x) = \begin{cases} \dfrac{2x^2 - 3x - 2}{x - 2}, & x \ne…Preview
  5. Q5Find whether the function is continuous or discontinuous at the indicated point: $f(x) = \begin{cases} \dfrac{|x - 4|}{2(x - 4)}, & x \ne 4…Preview
  6. Q6Find whether the function is continuous or discontinuous at the indicated point: $f(x) = \begin{cases} |x| \cos \dfrac{1}{x}, & x \ne 0 \\ 0…Preview
  7. Q7Find whether the function is continuous or discontinuous at the indicated point: $f(x) = \begin{cases} (x - a) \sin \dfrac{1}{x - a}, & x \n…Preview
  8. Q8Find whether the function is continuous or discontinuous at the indicated point: $f(x) = \begin{cases} \dfrac{e^{1/x}}{1 + e^{1/x}}, & x \ne…Preview
  9. Q9Find whether the function is continuous or discontinuous at the indicated point: $f(x) = \begin{cases} \dfrac{x^2}{2}, & 0 \le x \le 1 \\ 2x…Preview
  10. Q10Find whether the function is continuous or discontinuous at the indicated point: $f(x) = |x| + |x - 1|$ at $x = 1$.Preview
  11. Q11Find the value of $k$ so that the function $f$ is continuous at the indicated point: $f(x) = \begin{cases} 3x - 8, & x \le 5 \\ 2k, & x > 5…Preview
  12. Q12Find the value of $k$ so that the function $f$ is continuous at the indicated point: $f(x) = \begin{cases} \dfrac{2^{x+2} - 16}{4^x - 16}, &…Preview
  13. Q13Find the value of $k$ so that the function $f$ is continuous at the indicated point: $f(x) = \begin{cases} \dfrac{\sqrt{1 + kx} - \sqrt{1 -…Preview
  14. Q14Find the value of $k$ so that the function $f$ is continuous at the indicated point: $f(x) = \begin{cases} \dfrac{1 - \cos kx}{x \sin x}, &…Preview
  15. Q15Prove that the function $f$ defined by $f(x) = \begin{cases} \dfrac{x}{|x| + 2x^2}, & x \ne 0 \\ k, & x = 0 \end{cases}$ remains discontinuo…Preview
  16. Q16Find the values of $a$ and $b$ such that the function $f$ defined by $f(x) = \begin{cases} \dfrac{x - 4}{|x - 4|} + a, & x < 4 \\ a + b, & x…Preview
  17. Q17Given the function $f(x) = \dfrac{1}{x + 2}$. Find the points of discontinuity of the composite function $y = f(f(x))$.Preview
  18. Q18Find all points of discontinuity of the function $f(t) = \dfrac{1}{t^2 + t - 2}$, where $t = \dfrac{1}{x - 1}$.Preview
  19. Q19Show that the function $f(x) = |\sin x + \cos x|$ is continuous at $x = \pi$.Preview
  20. Q20Examine the differentiability of $f$, where $f$ is defined by $f(x) = \begin{cases} x[x], & 0 \le x < 2 \\ (x - 1)x, & 2 \le x < 3 \end{case…Preview
  21. Q21Examine the differentiability of $f$, where $f$ is defined by $f(x) = \begin{cases} x^2 \sin \dfrac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{case…Preview
  22. Q22Examine the differentiability of $f$, where $f$ is defined by $f(x) = \begin{cases} 1 + x, & x \le 2 \\ 5 - x, & x > 2 \end{cases}$ at $x =…Preview
  23. Q23Show that $f(x) = |x - 5|$ is continuous but not differentiable at $x = 5$.Preview
  24. Q24A function $f : \mathbb{R} \to \mathbb{R}$ satisfies the equation $f(x + y) = f(x)\,f(y)$ for all $x, y \in \mathbb{R}$, $f(x) \ne 0$. Suppo…Preview
  25. Q25Differentiate w.r.t. $x$: $2^{\cos^2 x}$.Preview
  26. Q26Differentiate w.r.t. $x$: $\dfrac{8^x}{x^8}$.Preview
  27. Q27Differentiate w.r.t. $x$: $\log\left(x + \sqrt{x^2 + a}\right)$.Preview
  28. Q28Differentiate w.r.t. $x$: $\log\left[\log(\log x^5)\right]$.Preview
  29. Q29Differentiate w.r.t. $x$: $\sin\sqrt{x} + \cos^2\sqrt{x}$.Preview
  30. Q30Differentiate w.r.t. $x$: $\sin^n(ax^2 + bx + c)$.Preview
  31. Q31Differentiate w.r.t. $x$: $\cos\left(\tan\sqrt{x + 1}\right)$.Preview
  32. Q32Differentiate w.r.t. $x$: $\sin x^2 + \sin^2 x + \sin^2(x^2)$.Preview
  33. Q33Differentiate w.r.t. $x$: $\sin^{-1}\left(\dfrac{1}{\sqrt{x + 1}}\right)$.Preview
  34. Q34Differentiate w.r.t. $x$: $(\sin x)^{\cos x}$.Preview
  35. Q35Differentiate w.r.t. $x$: $\sin^m x \cdot \cos^n x$.Preview
  36. Q36Differentiate w.r.t. $x$: $(x + 1)^2 (x + 2)^3 (x + 3)^4$.Preview
  37. Q37Differentiate w.r.t. $x$: $\cos^{-1}\left(\dfrac{\sin x + \cos x}{\sqrt{2}}\right),\ -\dfrac{\pi}{4} < x < \dfrac{\pi}{4}$.Preview
  38. Q38Differentiate w.r.t. $x$: $\tan^{-1}\left(\sqrt{\dfrac{1 - \cos x}{1 + \cos x}}\right),\ -\dfrac{\pi}{4} < x < \dfrac{\pi}{4}$.Preview
  39. Q39Differentiate w.r.t. $x$: $\tan^{-1}(\sec x + \tan x),\ -\dfrac{\pi}{2} < x < \dfrac{\pi}{2}$.Preview
  40. Q40Differentiate w.r.t. $x$: $\tan^{-1}\left(\dfrac{a\cos x - b\sin x}{b\cos x + a\sin x}\right),\ -\dfrac{\pi}{2} < x < \dfrac{\pi}{2}$ and $\…Preview
  41. Q41Differentiate w.r.t. $x$: $\sec^{-1}\left(\dfrac{1}{4x^3 - 3x}\right),\ 0 < x < \dfrac{1}{\sqrt{2}}$.Preview
  42. Q42Differentiate w.r.t. $x$: $\tan^{-1}\left(\dfrac{3a^2 x - x^3}{a^3 - 3ax^2}\right),\ -\dfrac{1}{\sqrt{3}} < \dfrac{x}{a} < \dfrac{1}{\sqrt{3…Preview
  43. Q43Differentiate w.r.t. $x$: $\tan^{-1}\left(\dfrac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}}\right),\ -1 < x < 1,\ x \…Preview
  44. Q44Find $\dfrac{dy}{dx}$ of the function expressed in parametric form: $x = t + \dfrac{1}{t},\ y = t - \dfrac{1}{t}$.Preview
  45. Q45Find $\dfrac{dy}{dx}$ of the function expressed in parametric form: $x = e^{\theta}\left(\theta + \dfrac{1}{\theta}\right),\ y = e^{-\theta}…Preview
  46. Q46Find $\dfrac{dy}{dx}$ of the function expressed in parametric form: $x = 3\cos\theta - 2\cos^3\theta,\ y = 3\sin\theta - 2\sin^3\theta$.Preview
  47. Q47Find $\dfrac{dy}{dx}$ of the function expressed in parametric form: $\sin x = \dfrac{2t}{1 + t^2},\ \tan y = \dfrac{2t}{1 - t^2}$.Preview
  48. Q48Find $\dfrac{dy}{dx}$ of the function expressed in parametric form: $x = \dfrac{1 + \log t}{t^2},\ y = \dfrac{3 + 2\log t}{t}$.Preview
  49. Q49If $x = e^{\cos 2t}$ and $y = e^{\sin 2t}$, prove that $\dfrac{dy}{dx} = -\dfrac{y \log x}{x \log y}$.Preview
  50. Q50If $x = a\sin 2t\,(1 + \cos 2t)$ and $y = b\cos 2t\,(1 - \cos 2t)$, show that $\left.\dfrac{dy}{dx}\right|_{t = \frac{\pi}{4}} = \dfrac{b}{a…Preview
  51. Q51If $x = 3\sin t - \sin 3t$, $y = 3\cos t - \cos 3t$, find $\dfrac{dy}{dx}$ at $t = \dfrac{\pi}{3}$.Preview
  52. Q52Differentiate $\dfrac{x}{\sin x}$ w.r.t. $\sin x$.Preview
  53. Q53Differentiate $\tan^{-1}\left(\dfrac{\sqrt{1 + x^2} - 1}{x}\right)$ w.r.t. $\tan^{-1} x$, when $x \ne 0$.Preview
  54. Q54Find $\dfrac{dy}{dx}$ when $x$ and $y$ are connected by the relation: $\sin(xy) + \dfrac{x}{y} = x^2 - y$.Preview
  55. Q55Find $\dfrac{dy}{dx}$ when $x$ and $y$ are connected by the relation: $\sec(x + y) = xy$.Preview
  56. Q56Find $\dfrac{dy}{dx}$ when $x$ and $y$ are connected by the relation: $\tan^{-1}(x^2 + y^2) = a$.Preview
  57. Q57Find $\dfrac{dy}{dx}$ when $x$ and $y$ are connected by the relation: $(x^2 + y^2)^2 = xy$.Preview
  58. Q58If $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$, then show that $\dfrac{dy}{dx} \cdot \dfrac{dx}{dy} = 1$.Preview
  59. Q59If $x = e^{x/y}$, prove that $\dfrac{dy}{dx} = \dfrac{x - y}{x \log x}$.Preview
  60. Q60If $y^x = e^{y - x}$, prove that $\dfrac{dy}{dx} = \dfrac{(1 + \log y)^2}{\log y}$.Preview
  61. Q61If $y = (\cos x)^{(\cos x)^{(\cos x)^{\cdots\infty}}}$, show that $\dfrac{dy}{dx} = \dfrac{y^2 \tan x}{y \log \cos x - 1}$.Preview
  62. Q62If $x\sin(a + y) + \sin a \cos(a + y) = 0$, prove that $\dfrac{dy}{dx} = \dfrac{\sin^2(a + y)}{\sin a}$.Preview
  63. Q63If $\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y)$, prove that $\dfrac{dy}{dx} = \sqrt{\dfrac{1 - y^2}{1 - x^2}}$.Preview
  64. Q64If $y = \tan^{-1} x$, find $\dfrac{d^2 y}{dx^2}$ in terms of $y$ alone.Preview
  65. Q65Find the points on the curve $y = (\cos x - 1)$ in $[0, 2\pi]$, where the tangent is parallel to the $x$-axis.Preview
  66. Q66Find a point on the curve $y = (x - 3)^2$, where the tangent is parallel to the chord joining the points $(3, 0)$ and $(4, 1)$.Preview
  67. Q67Find the values of $p$ and $q$ so that $f(x) = \begin{cases} x^2 + 3x + p, & x \le 1 \\ qx + 2, & x > 1 \end{cases}$ is differentiable at $x…Preview
  68. Q68If $x^m \cdot y^n = (x + y)^{m + n}$, prove that (i) $\dfrac{dy}{dx} = \dfrac{y}{x}$ and (ii) $\dfrac{d^2 y}{dx^2} = 0$.Preview
  69. Q69If $x = \sin t$ and $y = \sin pt$, prove that $(1 - x^2)\dfrac{d^2 y}{dx^2} - x\dfrac{dy}{dx} + p^2 y = 0$.Preview
  70. Q70Find $\dfrac{dy}{dx}$, if $y = x^{\tan x} + \dfrac{\sqrt{x^2 + 1}}{2}$.Preview
  71. Q71An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is __________.Preview
  72. Q72Derivative of $x^2$ w.r.t. $x^3$ is __________.Preview
  73. Q73If $f(x) = |\cos x|$, then $f'\left(\dfrac{\pi}{4}\right) = $ __________.Preview
  74. Q74If $f(x) = |\cos x - \sin x|$, then $f'\left(\dfrac{\pi}{3}\right) = $ __________.Preview
  75. Q75For the curve $\sqrt{x} + \sqrt{y} = 1$, $\dfrac{dy}{dx}$ at $\left(\dfrac{1}{4}, \dfrac{1}{4}\right)$ is __________.Preview
  76. Q76State whether True or False: If $f$ is continuous on its domain $D$, then $|f|$ is also continuous on $D$.Preview
  77. Q77State whether True or False: The composition of two continuous functions is a continuous function.Preview
  78. Q78State whether True or False: Trigonometric and inverse-trigonometric functions are differentiable in their respective domains.Preview
  79. Q79State whether True or False: If $f \cdot g$ is continuous at $x = a$, then $f$ and $g$ are separately continuous at $x = a$.Preview
  80. Q80If $f(x) = 2x$ and $g(x) = \dfrac{x^2}{2} + 1$, then which of the following can be a discontinuous function? (A) $f(x) + g(x)$ (B) $f(x) - g…Preview
  81. Q81The function $f(x) = \dfrac{4 - x^2}{4x - x^3}$ is (A) discontinuous at only one point (B) discontinuous at exactly two points (C) discontin…Preview
  82. Q82The set of points where the function $f$ given by $f(x) = |2x - 1|\sin x$ is differentiable is (A) $\mathbb{R}$ (B) $\mathbb{R} - \left\{\df…Preview
  83. Q83The function $f(x) = \cot x$ is discontinuous on the set (A) $\{x = n\pi : n \in \mathbb{Z}\}$ (B) $\{x = 2n\pi : n \in \mathbb{Z}\}$ (C) $\…Preview
  84. Q84The function $f(x) = e^{|x|}$ is (A) continuous everywhere but not differentiable at $x = 0$ (B) continuous and differentiable everywhere (C…Preview
  85. Q85If $f(x) = x^2 \sin \dfrac{1}{x}$, where $x \ne 0$, then the value of the function $f$ at $x = 0$, so that the function is continuous at $x…Preview
  86. Q86If $f(x) = \begin{cases} mx + 1, & x \le \dfrac{\pi}{2} \\ \sin x + n, & x > \dfrac{\pi}{2} \end{cases}$ is continuous at $x = \dfrac{\pi}{2…Preview
  87. Q87Let $f(x) = |\sin x|$. Then (A) $f$ is everywhere differentiable (B) $f$ is everywhere continuous but not differentiable at $x = n\pi$, $n \…Preview
  88. Q88If $y = \log\left(\dfrac{1 - x^2}{1 + x^2}\right)$, then $\dfrac{dy}{dx}$ is equal to (A) $\dfrac{4x^3}{1 - x^4}$ (B) $\dfrac{-4x}{1 - x^4}$…Preview
  89. Q89If $y = \sqrt{\sin x + y}$, then $\dfrac{dy}{dx}$ is equal to (A) $\dfrac{\cos x}{2y - 1}$ (B) $\dfrac{\cos x}{1 - 2y}$ (C) $\dfrac{\sin x}{…Preview
  90. Q90The derivative of $\cos^{-1}(2x^2 - 1)$ w.r.t. $\cos^{-1} x$ is (A) $2$ (B) $\dfrac{-1}{2\sqrt{1 - x^2}}$ (C) $\dfrac{2}{x}$ (D) $1 - x^2$Preview
  91. Q91If $x = t^2$, $y = t^3$, then $\dfrac{d^2 y}{dx^2}$ is (A) $\dfrac{3}{2}$ (B) $\dfrac{3}{4t}$ (C) $\dfrac{3}{2t}$ (D) $\dfrac{3}{4}$Preview

CBSE Sample Papers

Questions from official CBSE sample papers.

+Show 10 questions10 questions
  1. Q1If a function defined by 𝑓(𝑥) = { 𝑘𝑥 + 1, 𝑥 ≤ 𝜋 cos 𝑥 , 𝑥 > 𝜋 is continuous at 𝑥 = 𝜋, then the value of 𝑘 is (A) 𝜋 (B) −1 𝜋 (C) 0 (D) −2 𝜋Preview
  2. Q2If 𝑓(𝑥) = 𝑥 tan−1 𝑥 , then 𝑓′(1)is equal to (A) 𝜋 4 − 1 2 (B) 𝜋 4 + 1 2 (C) − 𝜋 4 − 1 2 (D) − 𝜋 4 + 1 2Preview
  3. Q3The interval in which the function $f$ defined by $f(x)=e^{x}$ is strictly increasing, is (A) $[1,\infty)$ (B) $(-\infty,0)$ (C) $(-\infty,\…Preview
  4. Q4The function $f:R\to Z$ defined by $f(x)=[x]$; where $[\,.\,]$ denotes the greatest integer function, is (A) Continuous at $x=2.5$ but not d…Preview
  5. Q5(C) cos−1(3𝑥) (D) 3 cos−1 𝑥 1 hPreview
  6. Q6(b) If $(x - a)^2 + (y - b)^2 = c^2$, for some $c > 0$, prove that $\dfrac{\left[1 + \left(\dfrac{dy}{dx}\right)^2\right]^{3/2}}{\dfrac{d^2y…Preview
  7. Q7The equation of the path traced by a roller-coaster is given by the polynomial $f(x) = a(x + 9)(x + 1)(x - 3)$. If the roller-coaster crosse…Preview
  8. Q8Differentiate $\tan^{-1}\left(\dfrac{1 + \cos x}{\sin x}\right)$ with respect to $x$.Preview
  9. Q9If $(x^2 + y^2)^2 = xy$, find $\dfrac{dy}{dx}$. **OR** If $x = a(2\theta - \sin 2\theta)$ and $y = a(1 - \cos 2\theta)$, find $\dfrac{dy}{dx…Preview
  10. Q10If $y = \sin(\sin x)$, prove that $\dfrac{d^2 y}{dx^2} + \tan x\, \dfrac{dy}{dx} + y\cos^2 x = 0$.Preview