If $x = \phi \left( t \right),y = {\rm{\psi }}\left( t \right),$ then $\frac{{{d^2}y}}{{d{x^2}}}$ is equal to

$\frac{{\phi '{\rm{\psi ''}} - {\rm{\psi '}}\phi ''}}{{{{\left( {\phi '} \right)}^2}}}$

$\frac{{\phi ''}}{{{\rm{\psi ''}}}}$

If $ y=\left(1+\frac{1}{x}\right)^{x} $, then $ \frac{d y}{d x} $ is equal to

$ \left(1+\frac{1}{x}\right)^{x}\left[\log \left(1+\frac{1}{x}\right)\right] $

$ \left(x+\frac{1}{x}\right)^{x}\left[\log (x-1)-\frac{x}{x+1}\right] $

If $f(x-y), f(x) f(y)$ and $f(x+y)$ are in A.P. for all $x, y \in R$ and $f(0)=0$, then

$f^{\prime}(-3)+f^{\prime}(3)=0$

Back to DIFFERENTIATION

Your Progress

0 / 932 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Mixed difficulty

Try a question, then open “View answer” to check it. Sign in to submit answers and save progress.

Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

Take a chapter quiz →

Showing 1-30 of 932 questions

1 Easy

1 of 932

Application +4 | -1

f(x)=|x|^{|\tan x|}

, then

f^{\prime}\left(\frac{-\pi}{6}\right)

is equal to

View answer and explanation

Correct option: B

\left(\frac{\pi}{6}\right)^{1 / \sqrt{3}}\left\{\frac{-2 \sqrt{3}}{\pi}+\frac{4}{3} \log \frac{6}{\pi}\right\}

Option B is correct because it is the result of differentiating the given function and evaluating at the specified point, considering the absolute value.

2 Easy

2 of 932

Application +4 | -1

Let

x

and

y

be two real numbers such that

x>0

and

x y=1

. The minimum value of

x+y

View answer and explanation

Correct option: C

Option C (2) is correct because the minimum value of x+y, given xy=1 and x>0, is 2, which can be found using calculus or AM-GM inequality.

3 Medium

3 of 932

Analyzing +4 | -1

Let

f(x)=[x] \sin (\pi x)

where

[x]

denotes the integer part of

x

. Then at

x=k, k

being an integer, the left derivative of

f(x)

View answer and explanation

Correct option: A

(-1)^k(k-1) \pi

Option A (

(-1)^k(k-1) \pi

) is correct because it is the result of calculating the left derivative of the function at x=k using the limit definition.

4 Easy

4 of 932

Analyzing +4 | -1

y = {\sec ^{ - 1}}\left[ {\mathrm{cosec}x} \right] + \mathrm{cose}{\mathrm{c}^{ - 1}}\left[ {\sec x} \right] + {\sin ^{ - 1}}[\cos x] + {\cos ^{ - 1}}[\sin x],

then

\frac{{dy}}{{dx}}

is equal to

View answer and explanation

Correct option: D

- 4

Option D (-4) is correct because after simplifying the expression using trigonometric identities, the derivative is found to be -4.

5 Easy

5 of 932

Application +4 | -1

\frac{d}{d x} \sin ^{-1}\left(3 x-4 x^{3}\right)

is equal to

View answer and explanation

Correct option: A

\frac{3}{\sqrt{1-x^{2}}}

The correct answer is A (

\frac{3}{\sqrt{1-x^{2}}}

) because it is the derivative of the simplified form of the given function after using trigonometric substitution.

6 Easy

6 of 932

Application +4 | -1

z = y + f\left( v \right),

where

v = \left( {\frac{x}{y}} \right),

then

v\frac{{\partial z}}{{\partial x}} + \frac{{\partial z}}{{\partial y}}

View answer and explanation

Correct option: B

Option B (1) is correct because after partial differentiation and substitution, the terms involving the function f' cancel out, leaving 1.

7 Easy

7 of 932

Application +4 | -1

The minimum value of

f(x)=|3-x|+

|2+x|+|5-x|

View answer and explanation

Correct option: B

Option B (7) is correct because it represents the minimum value of the given function, which occurs at x=3.

8 Medium

8 of 932

Application +4 | -1

Let

f(x)=\log \sin x^2, 0 < x < \pi / 2

then

f^{\prime \prime}(\sqrt{\pi} / 2)

is equal to

View answer and explanation

Correct option: C

2(1-\pi)

Option C (

2(1-\pi)

) is correct because it is the value of the second derivative of the given function evaluated at

x = \sqrt{\pi}/2

9 Medium

9 of 932

Analyzing +4 | -1

Let

f

be any function defined on

R

and let it satisfy the condition

\left| {f(x) - f(y)\left| {{\mkern 1mu} \le {\mkern 1mu} } \right|{{\left( {x - y} \right)}^2}} \right|,{\mkern 1mu} {\mkern 1mu} \forall \;x,y \in R

f(0) = 1

, then

View answer and explanation

Correct option: D

f(x) > 0,\,\,\forall x \in R

Option D is correct. Given the condition on

f(x)

, it can be shown that

f(x)

is a constant function, and since

f(0) = 1

f(x) = 1

for all

x

, thus

f(x) > 0

for all

x \in R

10 Medium

10 of 932

Analyzing +4 | -1

A function

f: R \rightarrow R

is differentiable and satisfies the equation

f\left(\frac{1}{n}\right)=0

for all integers

n \geq 1

, then

View answer and explanation

Correct option: B

f(0)=0=f^{\prime}(0)

Option B (

f(0)=0=f^{\prime}(0)

) is correct because the differentiability of

f

x=0

along with the given condition

f(1/n) = 0

forces both

f(0)

and

f'(0)

to be zero.

11 Easy

11 of 932

Application +4 | -1

x = \phi \left( t \right),y = \mathrm{\psi }\left( t \right),

then

\frac{{{d^2}y}}{{d{x^2}}}

is equal to

View answer and explanation

Correct option: B

\frac{{\phi '\psi '' - \psi '\phi ''}}{{{{\left( {\phi '} \right)}^3}}}

Option B (

\frac{{\phi '\psi '' - \psi '\phi ''}}{{{{\left( {\phi '} \right)}^3}}})

is correct because it represents the second derivative of y with respect to x when both x and y are parameterized by t.

12 Medium

12 of 932

Application +4 | -1

f(x)=(2 x-3 \pi)^{25}+\frac{4}{3} x+\cos x

and

g(x)

is the inverse of

f(x)

, then find

\frac{d}{d x}(g(x))

x=2 \pi

View answer and explanation

Correct option: B

\frac{3}{7}

Option B (3/7) is correct because it is the derivative of the inverse function g(x) evaluated at x=2*pi, which is the reciprocal of the derivative of f(x) evaluated at f^-1(2*pi).

13 Easy

13 of 932

Analyzing +4 | -1

f\left( x \right) = \frac{1}{{1 - x}},

then the derivative of the composite function

f\left[ {f\left\{ {f\left( x \right)} \right\}} \right]

is equal to

View answer and explanation

Correct option: C

Option C (1) is correct because after simplifying the composite function f[f{f(x)}] it results in x, and the derivative of x is 1.

14 Easy

14 of 932

Application +4 | -1

If for

x \in\left(0, \frac{1}{4}\right)

, the derivative of

\tan ^{-1}\left(\frac{6 x \sqrt{x}}{1-9 x^3}\right)

\sqrt{x} \cdot g(x)

, then

g(x)

equals

View answer and explanation

Correct option: B

\frac{9}{1+9 x^3}

Option B (9/(1+9x³)) is correct because after simplifying the inverse tangent expression and differentiating, the resulting g(x) is 9/(1+9x³).

15 Medium

15 of 932

Application +4 | -1

y=\frac{e^{2 x} \cos x}{x \sin x}

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: A

\frac{e^{2 x}\left[(2 x-1) \cot x-x \operatorname{cosec}^{2} x\right]}{x^{2}}

Option A is correct because the derivative is found using logarithmic differentiation and simplifying the resulting expression.

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

16 Hard

16 of 932

Application +4 | -1

The value of derivative of

\left| {x - 1} \right| + \left| {x - 3} \right|

x = 2

, is

View answer and explanation

Correct option: B

Option B (0) is correct because at x=2, the function is constant, and the derivative of a constant is zero.

17 Medium

17 of 932

Analyzing +4 | -1

y=y(x)

and it follows the relation

x \cos y+y \cos x=\pi

then

y^{\prime \prime}(0)=

View answer and explanation

Correct option: C

\pi

Option C (

\pi

) is correct because after differentiating the given equation twice and substituting x=0, we find that y''(0) =

\pi

18 Easy

18 of 932

Application +4 | -1

y=\left(1+\frac{1}{x}\right)^{x}

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: A

\left(1+\frac{1}{x}\right)^{x}\left[\log \left(1+\frac{1}{x}\right)-\frac{1}{1+x}\right]

Option A (

\left(1+\frac{1}{x}\right)^{x}\left[\log \left(1+\frac{1}{x}\right)-\frac{1}{1+x}\right]

) is correct because it is the result of logarithmic differentiation.

19 Easy

19 of 932

Application +4 | -1

The rate at which fluid level inside vertical cylindrical tank of radius

r

drop if we pump fluid out at the rate of

3 \mathrm{~cm}^3 \mathrm{l} / \mathrm{min}

View answer and explanation

Correct option: B

\frac{3}{\pi r^2}

Option B (

\frac{3}{\pi r^2}

) is correct because it represents the rate of change of height with respect to time when fluid is pumped out at a given rate.

20 Easy

20 of 932

Application +4 | -1

For

\left| x \right| < 1,

let

y = 1 + x + {x^2} + \ldots

\infty ,

then

\frac{{dy}}{{dx}} - y

is equal to

View answer and explanation

Correct option: D

x{y^2}

Option D (

x{y^2}

) is correct because, after finding the derivative

\frac{{dy}}{{dx}}

and simplifying, the expression

\frac{{dy}}{{dx}} - y

is equal to

x{y^2}

21 Easy

21 of 932

Analyzing +4 | -1

y = 1 - x + \frac{{{x^2}}}{{2!}} - \frac{{{x^3}}}{{3!}} + \frac{{{x^4}}}{{4!}} - ...,

then

\frac{{{d^2}y}}{{d{x^2}}}

is equal to

View answer and explanation

Correct option: D

y

The correct answer is D (

y

) because the second derivative of the given series is equal to the original series itself.

22 Easy

22 of 932

Application +4 | -1

f(x)=x^2+\tan x

and

f

is inverse of

g

, then

g^{\prime}(x)

is equal to

View answer and explanation

Correct option: D

\frac{1}{2 g(x)+1+\left(x-(g(x))^2\right)^2}

Option D is correct because it uses the inverse function derivative formula and correctly substitutes the derivative of the original function.

23 Medium

23 of 932

Application +4 | -1

The derivative of

\left[ {\frac{{{e^x} + 1}}{{{e^x}}}} \right]

is equal to

View answer and explanation

Correct option: C

- \frac{1}{{{e^x}}}

Option C (

- \frac{1}{{{e^x}}}

) is correct because it is the derivative of the given function after simplification.

24 Easy

24 of 932

Analyzing +4 | -1

f(x-y), f(x) f(y)

and

f(x+y)

are in A.P. for all

x, y \in R

and

f(0)=0

, then

View answer and explanation

Correct option: A

f^{\prime}(-2)=f^{\prime}(2)

Option A (

f^{\prime}(-2)=f^{\prime}(2)

) is correct because the given condition implies that f(x) is an odd function, which means its derivative f'(x) is an even function.

25 Medium

25 of 932

Application +4 | -1

f\left( x \right) = 10\cos x + \left( {13 + 2x} \right)\sin x

, then

f''\left( x \right) + f\left( x \right) =

View answer and explanation

Correct option: B

4\cos x

Option B (

4\cos x

) is correct because after finding the second derivative and adding it to the original function, the expression simplifies to

4\cos x

26 Easy

26 of 932

Analyzing +4 | -1

y = x\log \left( {\frac{x}{{a + bx}}} \right),

then

\frac{{{x^3}{d^2}y}}{{d{x^2}}}

is equal to

View answer and explanation

Correct option: B

{\left( {x\frac{{dy}}{{dx}} - y} \right)^2}

The correct answer is B (

{\left( {x\frac{{dy}}{{dx}} - y} \right)^2}

) because it is the result of differentiating the given function and substituting to find the expression for

{x^3}\frac{{{d^2}y}}{{d{x^2}}}

27 Easy

27 of 932

Application +4 | -1

f\left( x \right) = {\log _{{x^2}}}\left( {lo{g_e}x} \right),

then

f'\left( x \right)

x = e

View answer and explanation

Correct option: C

\frac{1}{{2e}}

Option C (

\frac{1}{{2e}}

) is correct because after differentiating and substituting x=e, the result is

\frac{1}{{2e}}

28 Easy

28 of 932

Analyzing +4 | -1

Let the functions

f:R \to R

and

g:R \to R

be defined as

f(x) = \left\{ {\begin{array}{cccccccccccccccccccc} {x + 2,}&{x\,\, < \,\,0}\\ {{x^2},\,}&{x\,\, \ge \,\,0} \end{array}} \right.\,

and

g(x) = \left\{ {\begin{array}{cccccccccccccccccccc} {{x^3},}&{x\,\, < \,\,1}\\ {3x - 2,}&{x\,\, \ge \,\,1} \end{array}} \right.\,

Then, the number of points in

R

where

(fog)\,(x)

is NOT differentiable is equal to :

View answer and explanation

Correct option: A

Option A (1) is correct because the function (fog)(x) is non-differentiable at only one point, x = 0, due to discontinuity.

29 Easy

29 of 932

Understanding +4 | -1

y = {\sin ^{ - 1}}\frac{x}{2} + {\cos ^{ - 1}}\frac{x}{2}

, then the value of

\frac{{dy}}{{dx}}

View answer and explanation

Correct option: C

Option C (0) is correct because the expression simplifies to a constant, and the derivative of a constant is zero.

30 Easy

30 of 932

Application +4 | -1

f(t)=\frac{t^2-5 t-1}{t^3}

then

f^{\prime}\left(\frac{1}{a}\right)

is given by

View answer and explanation

Correct option: B

3 a^4+10 a^3-a^2

Option B (

3 a^4+10 a^3-a^2

) is correct because it is the result of correctly differentiating the function and substituting

t = \frac{1}{a}

Revision notes and key takeaways

Review DIFFERENTIATION in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 932 single-correct-answer MCQs on DIFFERENTIATION.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

Showing 1-30 of 932 questions

Continue Your Practice

📚 Switch Set (ALL)

Chapter Sets

ALL 932 MCQs EASY 500 MCQs HARD 30 MCQs MEDIUM 402 MCQs

0 /932

0✓ 0✗

0% Accuracy