If $y=x+e^x$, then $\frac{d^2 x}{d y^2}$ is equal to

$\frac{1}{\left(1+e^x\right)^2}$

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

The derivative of $\log \left| x \right|$ is

$\frac{1}{{\left| x \right|}},x \ne 0$

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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Easy

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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 121-150 of 500 questions

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121 Easy

121 of 500

Application +4 | -1

f(x)=x^{2}-x+5, x>\frac{1}{2}

, and

\mathrm{g}(\mathrm{x})

is its inverse function, then

\mathrm{g}^{\prime}(7)

equals:

View answer and explanation

Correct option: C

\frac{1}{3}

Option C (

\frac{1}{3}

) is correct because it is the value of the derivative of the inverse function at x=7.

122 Easy

122 of 500

Application +4 | -1

for

0 < x < 2,\frac{d}{{dx}}\left( {{{\tan }^{ - 1}}\sqrt {\frac{{1 + \cos \frac{x}{2}}}{{1 - \cos \frac{x}{2}}}} } \right)

is equal to

View answer and explanation

Correct option: A

\frac{{ - 1}}{4}

Option A (

\frac{{ - 1}}{4}

) is correct because after simplifying the expression inside the arctangent function, the derivative is a constant.

123 Easy

123 of 500

Analyzing +4 | -1

y=x+e^x

, then

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

is equal to

View answer and explanation

Correct option: C

-\frac{e^x}{\left(1+e^x\right)^3}

Option C (-

\frac{e^x}{(1+e^x)^3}

) is correct because it is the result of finding the second derivative of the inverse function x(y).

124 Easy

124 of 500

Application +4 | -1

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

for

0 < x < 1

, where

[x]=

G.I.F., then

f^{\prime}\left(\frac{\sqrt{\pi}}{2}\right)

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because evaluating the derivative of the function at

x = \frac{{\sqrt{{\pi}}}}{{2}}

yields this result.

125 Easy

125 of 500

Application +4 | -1

f(x)=\log |x|, x \neq 0

then

f^{\prime}(x)

equals

View answer and explanation

Correct option: B

\frac{1}{x}

Option B is correct because the derivative of log|x| is 1/x for all x not equal to 0.

126 Easy

126 of 500

Application +4 | -1

The differential coefficient of

\cos ^{-1}\left\{\sqrt{\frac{1+x}{2}}\right\}

with respect to

x

is:

View answer and explanation

Correct option: A

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

Option A (-1/(2 sqrt(1-x²))) is correct because it is the derivative of the given function after simplification using trigonometric identities.

127 Easy

127 of 500

Analyzing +4 | -1

{\cos ^{ - 1}}\left( {\frac{y}{2}} \right) = {\log _e}{\left( {\frac{x}{5}} \right)^5},\left| y \right| < 2,

then:

View answer and explanation

Correct option: D

{x^2}y'' + xy' + 25y = 0

Option D (

{x^2}y'' + xy' + 25y = 0

) is correct because after differentiating the given equation twice and simplifying, it satisfies this relation.

128 Easy

128 of 500

Analyzing +4 | -1

y = \frac{{\sqrt {1 - \sin x} + \sqrt {1 + \sin x} }}{{\sqrt {1 - \sin x} - \sqrt {1 + \sin x} }},

then

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

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}\mathrm{cose}{\mathrm{c}^2}\frac{x}{2}

Option A (

\frac{1}{2}\mathrm{cose}{\mathrm{c}^2}\frac{x}{2}

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

\frac{1}{2}\mathrm{cose}{\mathrm{c}^2}\frac{x}{2}

129 Easy

129 of 500

Application +4 | -1

f\left( {x,y} \right) = \frac{{\cos (x - 4y)}}{{\cos \left( {x + 4y} \right)}},

then

{\left. {\frac{{\partial f}}{{\partial x}}} \right|_{y = \frac{\pi }{2}}}

is equal to

View answer and explanation

Correct option: B

Option B (0) is correct because when y = π/2, the function simplifies to 1, and its derivative with respect to x is 0.

130 Easy

130 of 500

Understanding +4 | -1

y = a{e^x} + b{e^{ - x}} + c

where

a,b,c

are parameters, then

y'

is equal to

View answer and explanation

Correct option: B

y'

Option B (

y'

) is correct because the third derivative of

y

is equal to the first derivative

y'

131 Easy

131 of 500

Application +4 | -1

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

then

f'\left( 1 \right)

is equal to

View answer and explanation

Correct option: A

\frac{1}{2} + \frac{\pi }{4}

Option A (

\frac{1}{2} + \frac{\pi }{4}

) is correct because after finding the derivative of

f(x) = x\tan^{-1}x

and substituting

x = 1

, we get

f'(1) = \frac{1}{2} + \frac{\pi }{4}

132 Easy

132 of 500

Application +4 | -1

y = \sqrt {x{{\log }_e}x,}

, then

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

x = e

View answer and explanation

Correct option: B

\frac{1}{{\sqrt e }}

Option B (

\frac{1}{{\sqrt e }}

) is correct because it is the value of the derivative of the function

y = \sqrt {x{{\log }_e}x}

evaluated at

x = e

133 Easy

133 of 500

Application +4 | -1

y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\ldots+\infty}}}

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: A

\frac{\cos x}{2 y-1}

Option A (

\frac{\cos x}{2 y-1}

) is correct because it is the derivative of the given function

y

found by recognizing the infinite nested radical and using implicit differentiation.

134 Easy

134 of 500

Application +4 | -1

y = \frac{{1 + {x^2} + {x^4}}}{{1 + x + {x^2}}}\mathrm{ \,\,and \,\,}\frac{{dy}}{{dx}} = ax + b,

then

View answer and explanation

Correct option: C

a = 2,\,\,b = - 1

Option C (

a = 2,\,\,b = - 1

) is correct because after simplifying the given expression and differentiating, the coefficients of x and the constant term match these values.

135 Easy

135 of 500

Understanding +4 | -1

The derivative of

\log \left| x \right|

View answer and explanation

Correct option: C

\frac{1}{x},x \ne 0

Option C (

\frac{1}{x}, x \ne 0

) is correct because the derivative of

\log |x|

\frac{1}{x}

for both

x > 0

and

x < 0

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136 Easy

136 of 500

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} \,g(x)

, then

g(x)

equals

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is the result of differentiating the given function and simplifying the expression.

137 Easy

137 of 500

Application +4 | -1

f(x)=x^n+4

then the value of

f(1)+

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

View answer and explanation

Correct option: B

2^n+4

Option B (

2^n+4

) is correct because the expression is a Taylor series expansion of

f(x)

around

x=1

, which evaluates to

f(2) = 2^n + 4

138 Easy

138 of 500

Application +4 | -1

y=a \sin m x+b \cos m x

, then

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

is equal to

View answer and explanation

Correct option: B

-m^2 y

Option B (

-m^2 y

) is correct because differentiating the given equation twice leads to this result.

139 Easy

139 of 500

Application +4 | -1

Let

f(x)=\sin x, g(x)=x^2

and

h(x)=\log _e x

. If

F(x)=(\operatorname{hog of})(x)

, then

F^{\prime \prime}(x)

is equal to

View answer and explanation

Correct option: D

-2 \operatorname{cosec}^2 x

Option D (

-2 \operatorname{cosec}^2 x

) is correct because it is the second derivative of the composite function

F(x) = (h \circ g \circ f)(x)

140 Easy

140 of 500

Analyzing +4 | -1

y = \mathrm{In}{\left( {\frac{x}{{a + bx}}} \right)^x},

then

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

is equal to

View answer and explanation

Correct option: D

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

The correct answer is D (

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

) because it is derived through logarithmic differentiation and algebraic manipulation.

141 Easy

141 of 500

Application +4 | -1

y=\sum\limits_{r=1}^x \tan ^{-1}\left(\frac{1}{1+r+r^2}\right)

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: B

\frac{1}{1+(1+x)^2}

Option B (

\frac{1}{1+(1+x)^2}

) is correct because the summation simplifies to a telescoping series, and the derivative is then straightforward.

142 Easy

142 of 500

Application +4 | -1

The value of

a

in order that

f(x)=\sqrt{3} \sin x-\cos x-2 a x+b

decreases for all real value of

x

, is given by

View answer and explanation

Correct option: B

a \geq 1

Option B (

a \geq 1

) is correct because for the function to decrease for all real values of x, its derivative must be negative, leading to the inequality

a \geq 1

143 Easy

143 of 500

Application +4 | -1

Derivative of the function

f(x)=\log _{5}\left(\log _{7} x\right), x > 7

View answer and explanation

Correct option: A

\frac{1}{x(\ln 5)(\ln 7)\left(\log _{7} x\right)}

Option A (

\frac{1}{x(\ln 5)(\ln 7)(\log _{7} x)}

) is correct because it is the result of applying the chain rule and change of base formula for logarithms.

144 Easy

144 of 500

Analyzing +4 | -1

The second order derivative of

a \sin ^3 t

with respect to a

\cos ^3 t

t=\pi / 4

, is

View answer and explanation

Correct option: A

\frac{4 \sqrt{2}}{3 a}

Option A (

\frac{4 \sqrt{2}}{3 a}

) is correct because after finding the first and second derivatives and substituting t =

\pi/4

, the result is

\frac{4 \sqrt{2}}{3 a}

145 Easy

145 of 500

Application +4 | -1

y = {x^n}\log x + x{\left( {\log x} \right)^n},

then

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

is equal to

View answer and explanation

Correct option: A

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

The correct answer is A. The derivative is found by applying the product rule and chain rule to the given function.

146 Easy

146 of 500

Application +4 | -1

Let

g

be the inverse function of differentiable function

f

and

f'(x) = \frac{{{x^{10}}}}{{1 + {x^2}}}.

g(2) = a,\,\,

then

g'(2)

is equal to

View answer and explanation

Correct option: B

\frac{{1 + {a^2}}}{{{a^{10}}}}

Option B (

\frac{{1 + {a^2}}}{{{a^{10}}}}

\) is correct because it is the value of the derivative of the inverse function g at 2, calculated using the derivative of the original function f.

147 Easy

147 of 500

Application +4 | -1

y=\log \sqrt{\tan x}

, then the value of

\frac{d y}{d x}

x=\frac{\pi}{4}

, is given by

View answer and explanation

Correct option: B

Option B (1) is correct because substituting x=

\frac{\pi}{4}

into the derivative of the function yields 1.

148 Easy

148 of 500

Application +4 | -1

f(x)=x^n

and

f^{\prime}(1)=10

, then the value of '

n

' is

View answer and explanation

Correct option: A

Option A (10) is correct because the value of 'n' is 10, derived from the derivative of the function at x=1.

149 Easy

149 of 500

Understanding +4 | -1

Which one of the following is not true always?

View answer and explanation

Correct option: B

f\left( x \right)

is continuous at

x = a

, then it is differentiable at

x = a

Option B (If

f(x)

is continuous at

x = a

, then it is differentiable at

x = a

) is correct because continuity does not guarantee differentiability.

150 Easy

150 of 500

Application +4 | -1

y = {\sin ^{ - 1}}\left( {\frac{{5x + 12\sqrt {1 - {x^2}} }}{{13}}} \right),

then

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

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because after substituting

x = \sin \theta

and simplifying, the derivative is

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

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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