If $y=\{f(x)\}^{\phi(x)}$, then $\frac{d y}{d x}$ is

$\frac{\phi}{f}\left(\frac{d f}{d x}\right)+\frac{d \phi}{d x} \log f$

$e^{\phi \log f}\left\{\phi \frac{f^{\prime}}{f}+\phi^{\prime} \log f^{\prime}\right\}$

Let $f:R \to R$ be differentiable at $c \in R$ and $f(c) = 0.$ If $g(x) = \left| {f(x)} \right|,$ then at $x = c,$ $g$ is

Differentiable if $f'(c) \ne 0$

Not differentiable if $f'(c) = 0$

If $\tan ^{-1} y-y+x=0$ then $\frac{d^2 y}{d x^2}$ is equal to

$\frac{2\left(1+y^2\right)}{y^4}$

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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Easy

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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.

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301 Easy

301 of 500

Application +4 | -1

f(a)=2, f^{\prime}(a)=1, g(a)=1, g^{\prime}(a)=2

, then

\lim _{x \rightarrow a} \frac{g(x) f(a)-g(a) f(x)}{x-a}

is equal to

View answer and explanation

Correct option: D

Option D (3) is correct because by applying L'Hôpital's Rule and using the given values of the functions and their derivatives, the limit evaluates to 3.

302 Easy

302 of 500

Application +4 | -1

f(3)=6

and

f^{\prime}(3)=2

, then

\lim\limits _{x \rightarrow 3} \frac{x f(3)-3 f(x)}{x-3}

is given by

View answer and explanation

Correct option: C

Option C (0) is correct because after manipulation and applying the definition of the derivative, the expression simplifies to 6 - 3*2 = 0.

303 Easy

303 of 500

Application +4 | -1

f(x)=x e^{-x^2 / 2}

, then

f^{\prime}(1)

View answer and explanation

Correct option: B

Option B is correct because it is the result of differentiating the function and evaluating the derivative at x=1.

304 Easy

304 of 500

Application +4 | -1

z = \log (\tan x + \tan y),

then

(\sin 2x)\frac{{\partial z}}{{\partial x}} + (\sin 2y)\frac{{\partial z}}{{\partial y}}

is equal to

View answer and explanation

Correct option: B

Option B (2) is correct because, after calculating the partial derivatives and substituting them into the given expression, the result simplifies to 2.

305 Easy

305 of 500

Analyzing +4 | -1

x = {e^{y + {e^{y + {e^{\,y + \,...\,\,{\text{to}}\,\infty }}}}}},

x > 0,

then

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

View answer and explanation

Correct option: C

\frac{1-x}{x}

Option C (

\frac{1-x}{x}

) is correct because it is derived by taking the logarithm of the given equation and then differentiating with respect to x.

306 Easy

306 of 500

Application +4 | -1

{\sin ^{ - 1}}\left( {\frac{{{x^2} - {y^2}}}{{{x^2} + {y^2}}}} \right) = \log a

, then

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

equals

View answer and explanation

Correct option: D

\frac{y}{x}

Option D (

\frac{y}{x}

) is correct because implicit differentiation leads to the relationship

\frac{dy}{dx} = \frac{y}{x}

307 Easy

307 of 500

Application +4 | -1

3 f(x)-2 f\left(\frac{1}{x}\right)=x

, then

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

View answer and explanation

Correct option: B

\frac{1}{2}

Option B (-a) is correct because the point P(a, f(a)) is nearest to (0,0) when the tangent at P is perpendicular to the line joining P and (0,0).

308 Easy

308 of 500

Application +4 | -1

f(x)=\cos \left(x^2-4[x]\right)

for

0 < x < 1

, where

[x]

denotes the greatest integer

\leq \mathrm{x}

, then

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

is equal

View answer and explanation

Correct option: A

-\sqrt{\frac{\pi}{2}}

Option A (

-\sqrt{\frac{\pi}{2}}

) is correct because the derivative of the function is calculated and then evaluated at the given point.

309 Easy

309 of 500

Analyzing +4 | -1

The derivative of

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

with respect to

\sqrt{1-x^2}

x=\frac{1}{2}

, is

View answer and explanation

Correct option: B

Option B (4) is correct because by substituting

x = cos(\theta)

, we can simplify the inverse trigonometric function and then find the derivative with respect to

\sqrt{1-x^2}

310 Easy

310 of 500

Analyzing +4 | -1

f(x)=\left|\begin{array}{ccc}x^n & \sin x & \cos x \\ n ! & \sin \frac{n \pi}{2} & \cos \frac{n \pi}{2} \\ a & a^2 & a^3\end{array}\right|

, then the value of

\frac{d^n}{d x^n}(f(x))

x=0

for

n=2 m+1

, is

View answer and explanation

Correct option: B

Option B (0) is correct because the nth derivative of f(x) at x=0 is a determinant with two identical rows, making the determinant equal to zero.

311 Easy

311 of 500

Analyzing +4 | -1

f\left( {x + y} \right) = 2f\left( x \right)f\left( y \right),f'\left( 5 \right) = 1024\left( {\log 2} \right)

and

f\left( 2 \right) = 8,

then the value of

f'\left( 3 \right)

View answer and explanation

Correct option: A

64\left( {\log 2} \right)

Option A (

64\left( {\log 2} \right)

) is correct because it is the value of

f'(3)

obtained after using the given functional equation and derivative information.

312 Easy

312 of 500

Application +4 | -1

z(t)=\sqrt{t^3+1}

, then

z^{\prime}(2)

is equal to

View answer and explanation

Correct option: C

Option C (2) is correct because it is the value of the derivative of the given function at t=2, calculated using the chain rule.

313 Easy

313 of 500

Application +4 | -1

y=\{f(x)\}^{\phi(x)}

, then

\frac{d y}{d x}

View answer and explanation

Correct option: A

f^{\phi \log f}\left\{\frac{\phi}{f} \frac{d f}{d x}+\log f \cdot \frac{d \phi}{d x}\right\}

Option A (

f^{\phi \log f}\left{\frac{{\phi}}{{f}} \frac{{d f}}{{d x}}+\log f \cdot \frac{{d \phi}}{{d x}}\right\}

) is correct because it is the result of applying logarithmic differentiation to the function

y = \{f(x)\} ^{\phi(x)}

314 Easy

314 of 500

Application +4 | -1

f\left( x \right) = {x^n} + 4

, then the value of

f\left( 1 \right) + \frac{{f'\left( 1 \right)}}{{1!}} + \frac{{f''\left( 1 \right)}}{{2!}} + ... + \frac{{{f^n}\left( 1 \right)}}{{n!}}

View answer and explanation

Correct option: B

{2^n} + 4

Option B (

{2^n} + 4

) is correct because it is the result of applying Taylor's theorem and the binomial theorem.

315 Easy

315 of 500

Application +4 | -1

For the curves

x=a \sqrt{\cos 2 \theta} \cos \theta, y=a \sqrt{\cos 2 \theta} \sin \theta

, the tangent at

\theta=\frac{\pi}{6}

is ......

View answer and explanation

Correct option: A

Parallel to

\mathrm{X}

-axis or

\mathrm{X}

-axis itself.

Option A (Parallel to X-axis or X-axis itself) is correct because at

\theta = \frac{\pi}{6}

, the derivative

\frac{dy}{dx}

is 0, indicating a horizontal tangent.

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316 Easy

316 of 500

Application +4 | -1

Let

f

and

g

be differentiable functions satisfying

g^{\prime}(a)=2, g(a)=b

and

fog = I

(identity function). Then,

f^{\prime}(b)

is equal to

View answer and explanation

Correct option: C

01-Feb

Option C (01-Feb), which translates to 1/2, is correct because by differentiating

f(g(x)) = x

and substituting the given values, we find

f'(b) = 1/2

317 Easy

317 of 500

Analyzing +4 | -1

The derivative of

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

with respect to

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

View answer and explanation

Correct option: B

Option B (1) is correct because both

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

and

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

are equal to

2{\tan ^{ - 1}}x

, so their derivatives with respect to each other is 1.

318 Easy

318 of 500

Application +4 | -1

f(x)=x^n

, then the value of

f(1)-\frac{f^{\prime}(1)}{1 !}+\frac{f^{\prime \prime}(1)}{2 !}-\frac{f^{\prime \prime \prime}(1)}{3 !}+\cdots+\frac{(-1)^n f^n(1)}{n !}

View answer and explanation

Correct option: B

Option B (0) is correct because the given expression is the binomial expansion of (1-1)^n.

319 Easy

319 of 500

Analyzing +4 | -1

y = \left( {1 + x} \right)\left( {1 + {x^2}} \right)\left( {1 + {x^4}} \right) \ldots \left( {1 + {x^{2n}}} \right),

then the value of

{\left( {\frac{{dy}}{{dx}}} \right)_{x = 0}}

View answer and explanation

Correct option: C

Option C (1) is correct because after expanding the product and differentiating, the value of the derivative at x=0 is 1.

320 Easy

320 of 500

Application +4 | -1

y=\sec \left(\tan ^{-1} x\right)

, then

\frac{d y}{d x}

x=1

is equal to:

View answer and explanation

Correct option: A

\frac{1}{\sqrt{2}}

Option A (

\frac{1}{\sqrt{2}}

) is correct because it is the value of the derivative of y with respect to x at x=1.

321 Easy

321 of 500

Application +4 | -1

f(x)=\tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right)

, then

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

is equal to

View answer and explanation

Correct option: C

\frac{3}{1+x^2}

for all

x \in R-\left\{-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right\}

Option C is correct because it correctly simplifies the inverse tangent function and finds its derivative, considering the domain restrictions.

322 Easy

322 of 500

Understanding +4 | -1

At how many points between the interval

(-\infty, \infty)

is the function

f(x)=\sin x

is not differentiable.

View answer and explanation

Correct option: A

Option A (0) is correct because the sine function, sin(x), is differentiable for all real numbers.

323 Easy

323 of 500

Application +4 | -1

Let

f: R \rightarrow R

be a function given by

f(x+y)=f(x) f(y)

for all

x, y \in R

f(x)=1+x g(x) \cdot \log _e 2

, where

\lim\limits _{x \rightarrow 0} g(x)=1

. Then,

f^{\prime}(x)=

View answer and explanation

Correct option: A

\log _e 2^{f(x)}

Option A is correct because using the given condition

f(x)=1+x g(x) \cdot \log _e 2

, where

\lim\limits _{x \rightarrow 0} g(x)=1

, and the functional equation, we find

f^{\prime}(x)=\log _e 2^{f(x)}

324 Easy

324 of 500

Application +4 | -1

If the function

f(x)=x^3+e^{x / 2}

and

g(x)=f^{-1}(x)

, then the value of

g^{\prime}(1)

View answer and explanation

Correct option: B

Option B (2) is correct because it uses the derivative of an inverse function and the given values.

325 Easy

325 of 500

Application +4 | -1

f\left( x \right) = {\log _e}\left\{ {\frac{{u\left( x \right)}}{{\upsilon \left( x \right)}}} \right\},u\left( 1 \right) = \upsilon \left( 1 \right)

and

u'\left( 1 \right) = \upsilon '\left( 1 \right) = 2

, then

f'\left( 1 \right)

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because when

u(1) = v(1)

and

u'(1) = v'(1) = 2

, the derivative

f'(1)

becomes zero.

326 Easy

326 of 500

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)

is equal to

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 terms simplify to

4\cos x

327 Easy

327 of 500

Analyzing +4 | -1

Let

f:R \to R

be differentiable at

c \in R

and

f(c) = 0.

g(x) = \left| {f(x)} \right|,

then at

x = c,

g

View answer and explanation

Correct option: D

Differentiable if

f'(c) = 0

Option D (Differentiable if

f'(c) = 0

) is correct because if

f'(c) = 0

, then the limit defining

g'(c)

exists and is equal to 0, making

g

differentiable.

328 Easy

328 of 500

Application +4 | -1

y = {e^x}.{e^{{x^2}}}.{e^{{x^3}}}{._.}...{e^{{x^n}}}...,

for

0 < x < 1

, then

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

x = \frac{1}{2}

View answer and explanation

Correct option: B

4e

Option B (

4e

) is correct because after simplifying the expression and differentiating, the derivative evaluated at

x = \frac{1}{2}

4e

329 Easy

329 of 500

Application +4 | -1

\tan ^{-1} y-y+x=0

then

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

is equal to

View answer and explanation

Correct option: A

\frac{-2\left(1+y^2\right)}{y^5}

Option A (

\frac{{-2(1+y^2)}}{{y^5}}

) is correct because it is the result of differentiating the given equation twice with respect to x.

330 Easy

330 of 500

Application +4 | -1

{2^x} + {2^y} = {2^{x + y}}

, then the value of

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

x = y = 1

, is

View answer and explanation

Correct option: B

- 1

Option B (-1) is correct because it is the value of dy/dx at x=y=1, obtained by implicit differentiation of the given equation.

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

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