If $y = {2^x}{.3^{2x - 1}},$ then $\frac{{{d^2}y}}{{d{x^2}}}$ is equal to

$\left( {\log 2} \right)\left( {\log 3} \right)$

If $x=f(t), y=g(t)$, then $\frac{d^2 y}{d x^2}=\ldots \ldots$

$\frac{f^{\prime} g^{\prime \prime}-g^{\prime} f^{\prime}}{(f)^2}$

$\frac{f^{\prime \prime}}{g^{\prime \prime}}$

$\frac{{g''}}{{{f^{\prime \prime }}}}$

If $ \cos (x+y)=y \sin x $, then $ \frac{d y}{d x} $ is equal to

$ \frac{y \cos x-\sin (x+y)}{\sin x-\sin (x+y)} $

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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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Showing 151-180 of 500 questions

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151 Easy

151 of 500

Application +4 | -1

A rectangle with perimeter

32 \mathrm{~cm}

has greatest area if its length is

View answer and explanation

Correct option: C

Option C (8) is correct because a rectangle with a fixed perimeter has the greatest area when it is a square.

152 Easy

152 of 500

Analyzing +4 | -1

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

, then

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

View answer and explanation

Correct option: B

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

153 Easy

153 of 500

Application +4 | -1

y = {\tan ^{ - 1}}\left( {\frac{{\cos x}}{{1 + \sin x}}} \right)

, then

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

is equal to

View answer and explanation

Correct option: D

- \frac{1}{2}

Option D is correct because the expression can be simplified using trigonometric identities, leading to a simple derivative.

154 Easy

154 of 500

Application +4 | -1

Let

f

be a one-one function satisfying

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

then

\left(f^{-1}\right)^{\prime \prime}(x)

is equal to

View answer and explanation

Correct option: B

-1 / x^2

Option B (

-1 / x^2

) is correct because it is the second derivative of the inverse function, calculated using the chain rule and the given condition.

155 Easy

155 of 500

Application +4 | -1

\frac{d}{{dx}}\sqrt {\frac{{1 - \sin 2x}}{{1 + \sin 2x}}}

is equal to

View answer and explanation

Correct option: B

- {\sec ^2}\left( {\frac{\pi }{4} - x} \right)

Option B (

\ - {\sec ^2}\left( {\frac{\pi }{4} - x} \right)

) is correct because it is the correct derivative of the simplified expression after using trigonometric identities.

156 Easy

156 of 500

Application +4 | -1

Let

f(x)=|x|

and

g(x)=[x]

where [.] denotes the greatest integer function. Then,

(f \circ g)^{\prime}(-2)

, is

View answer and explanation

Correct option: D

none of these

The correct answer is D (none of these) because the function f o g is not differentiable at x = -2.

157 Easy

157 of 500

Analyzing +4 | -1

The derivative of

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

with respect to

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

x=\frac{1}{2}

, is

View answer and explanation

Correct option: A

\frac{\sqrt{3}}{10}

Option A (

\frac{\sqrt{3}}{10}

) is correct because by substituting

x = tan(\theta)

and

x = sin(\phi)

, we can simplify both inverse trigonometric functions and then find the derivative at x=1/2.

158 Easy

158 of 500

Application +4 | -1

The derivative of

\log _{10} x

with respect to

x^2

, is

View answer and explanation

Correct option: B

\frac{2}{x^2} \log _{10} e

Option C is correct because it is derived by using the change of base formula and then differentiating with respect to x².

159 Easy

159 of 500

Application +4 | -1

The length of the subnormal of the curve

y^2=8 a x(a > 0)

is -

View answer and explanation

Correct option: B

4 \mathrm{a}

Option B (

4a

) is correct because the length of the subnormal is calculated using the derivative of the curve's equation.

160 Easy

160 of 500

Analyzing +4 | -1

f(x)=|x-1|

and

g(x)=f(f(f(x)))

, then for

x>2, g^{\prime}(x)

is equal to

View answer and explanation

Correct option: C

-1

for

2 < x < 3

Option C (

-1

for

2 < x < 3

) is correct because for

x > 2

g(x) = 3 - x

, and thus

g'(x) = -1

161 Easy

161 of 500

Application +4 | -1

y = {e^{\frac{1}{2}\log (1 + {{\tan }^2}x)}}

, then

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

is equal to

View answer and explanation

Correct option: C

\sec x\tan x

Option C (

\sec x\tan x

) is correct because, after simplifying the expression for y, the derivative

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

is found to be

\sec x\tan x

162 Easy

162 of 500

Application +4 | -1

f(x)=\sin \left(\frac{\pi}{3}[x]-x^2\right)

, where

2 < x < 3

and [.] represetns greatest integer function, then

f^{\prime}\left(\sqrt{\frac{5 \pi}{3}}\right)

is equal to

View answer and explanation

Correct option: B

2 \sqrt{\frac{5 \pi}{3}}

Option B (

2 \sqrt{\frac{5 \pi}{3}}

) is correct because by substituting the greatest integer function and differentiating

f(x)

, we can find

f'(\sqrt{\frac{5 \pi}{3}})

163 Easy

163 of 500

Application +4 | -1

f\left( x \right) = \left| {x - 2} \right| + \left| {x + 1} \right| - x,

then

f'\left( { - 10} \right)

is equal to

View answer and explanation

Correct option: A

- 3

Option A (-3) is correct because for x < -1, the function simplifies to f(x) = -3x + 1, and thus f'(x) = -3.

164 Easy

164 of 500

Application +4 | -1

Derivative of

\sin x

w.r.t.

\cos x

View answer and explanation

Correct option: C

- \cot x

Option C (

- \cot x

) is correct because it is the derivative of

\sin x

with respect to

\cos x

obtained using the chain rule.

165 Easy

165 of 500

Application +4 | -1

y = \sin \left( {m{{\sin }^{ - 1}}x} \right)

, then

\left( {1 - {x^2}} \right)\frac{{{d^2}y}}{{d{x^2}}} =

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the result of differentiating the given equation twice and rearranging the terms.

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166 Easy

166 of 500

Application +4 | -1

The differential coefficient of

f(\sin x)

with respect to

x

, where

f\left( x \right) = \log x

, is

View answer and explanation

Correct option: B

\cot x

Option B (

\cot x

) is correct because the derivative of

f(\sin x)

with respect to

x

, where

f(x) = \log x

, is indeed

\cot x

167 Easy

167 of 500

Analyzing +4 | -1

The function

f(x)

given by

f(x)=\,arc\,\,sin \left(\frac{2 x}{1+x^2}\right)

, is

View answer and explanation

Correct option: B

such that

f'(x) = \left\{ {\begin{array}{llllllllllllllllllll} {\frac{2}{{1 + {x^2}}},\,\,\,\, - 1 < x < 1}\\ {\frac{{ - 2}}{{1 + {x^2}}},\,\,\,\,|x| > 1} \end{array}} \right.

Option B is correct because the derivative of the arcsin function needs to be evaluated piecewise due to the range restriction of the arcsin function, resulting in different derivatives for |x| < 1 and |x| > 1.

168 Easy

168 of 500

Application +4 | -1

\mathrm{g}

is the inverse of

\mathrm{f}

and

\mathrm{f}^{\prime}(x)=\frac{1}{1+x^3}

, then find

g^{\prime}(x)

View answer and explanation

Correct option: C

1+[g(x)]^3

Option C (

1+[g(x)]^3

) is correct because by differentiating

f[g(x)] = x

with respect to x and using the given

f'(x)

, we find that

g'(x) = 1 + [g(x)]^3

169 Easy

169 of 500

Application +4 | -1

y = {2^x}{.3^{2x - 1}},

then

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

is equal to

View answer and explanation

Correct option: D

{(\log 18)^2}y

Option D (

{(\log 18)^2}y

) is correct because the second derivative of the given function can be expressed in terms of the original function and the square of the logarithm of 18.

170 Easy

170 of 500

Application +4 | -1

The differential coefficient of

f\left( {\log x} \right),

where

\,f\left( x \right) = \log x

View answer and explanation

Correct option: B

{(x\log x)^{ - 1}}

Option B (

{(x\log x)^{ - 1}}

) is correct because the derivative of

\log(\log x)

\frac{1}{{x\log x}}

171 Easy

171 of 500

Application +4 | -1

x=f(t), y=g(t)

, then

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

View answer and explanation

Correct option: B

\frac{{{f^\prime }g'' - {g^\prime }{f^{\prime \prime }}}}{{{{(f)}^3}}}

Option B is correct because it represents the correct formula for the second derivative of y with respect to x when both x and y are functions of t.

172 Easy

172 of 500

Application +4 | -1

f\left( x \right)

is a polynomial of degree 2,

f\left( 0 \right) = 4,f'\left( 0 \right) = 3

and

f''\left( 0 \right) = 4

, then

f\left( { - 1} \right)

is equal to

View answer and explanation

Correct option: A

Option A (3) is correct because given the conditions, the polynomial is determined to be

f(x) = 2x^2 + 3x + 4

, and

f(-1) = 3

173 Easy

173 of 500

Application +4 | -1

2{x^2} - 3xy + {y^2} + x + 2y - 8 = 0,

then

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

is equal to

View answer and explanation

Correct option: A

\frac{{3y - 4x - 1}}{{2y - 3x + 2}}

Option A (

\frac{{3y - 4x - 1}}{{2y - 3x + 2}}

) is correct because it is the result of implicit differentiation of the given equation with respect to x.

174 Easy

174 of 500

Application +4 | -1

Let

f: R \rightarrow R

and

g: R \rightarrow R

be differentiable functions such that

f(x)=x^3+3 x+2, g(f(x))=x

for all

x \in R

. Then,

g^{\prime}(2)=

View answer and explanation

Correct option: C

\frac{1}{3}

Option C (

\frac{{1}}{{3}}

) is correct because it is derived by differentiating

g(f(x)) = x

and substituting

x=0

175 Easy

175 of 500

Application +4 | -1

On the curve

y=x^3

, the point at which the tangent line in parallel to the chord through the point

(-1,-1)

and

(2,8)

View answer and explanation

Correct option: A

(1,1)

Option A ((1,1)) is correct because the slope of the tangent at that point is equal to the slope of the chord connecting (-1,-1) and (2,8).

176 Easy

176 of 500

Application +4 | -1

f(x)=(1+x)^n

, then the value of

f(0)+

f^{\prime}(0)+\frac{1}{2 !} f^{\prime \prime}(0)+\cdots+\frac{1}{n !} f^n(0)

View answer and explanation

Correct option: B

2 n

Option D (none of these) is correct because the expression simplifies to

2^n

, not

2n

177 Easy

177 of 500

Analyzing +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 !}+\ldots \ldots \ldots . . \frac{(-1)^{n} f^{n}(1)}{n !}

View answer and explanation

Correct option: D

0 .

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

178 Easy

178 of 500

Application +4 | -1

\cos (x+y)=y \sin x

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: A

-\frac{\sin (x+y)+y \cos x}{\sin x+\sin (x+y)}

Option A is correct because it is the result of implicit differentiation of the given equation and solving for dy/dx.

179 Easy

179 of 500

Application +4 | -1

\frac{d}{d x}\left[e^{-a x^{2}} \log (\sin x)\right]

View answer and explanation

Correct option: C

e^{-a x^{2}}(\cot x-2 a x \log \sin x)

Option C is correct because the derivative is found using the product rule and simplifying the resulting expression.

180 Easy

180 of 500

Analyzing +4 | -1

{x^2} + {y^2} = t + \frac{1}{t}

and

{x^4} + {\mathrm{y}^4} = {\mathrm{t}^2} + \frac{1}{{{\mathrm{t}^2}}},

then

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

is equal to

View answer and explanation

Correct option: B

- \frac{y}{x}

Option B (

\ - \frac{y}{x}

) is correct because by manipulating the given equations, we find that

{x^2}{y^2} = 1

, and differentiating this yields

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

Revision notes and key takeaways

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

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