If $ x \log _{e}\left(\log _{e} x\right)-x^{2}+y^{2}=4(y>0) $, then $ \frac{d y}{d x} $ at $ \mathrm{x}=\mathrm{e} $ is equal to :

$ \frac{(1+2 e)}{2 \sqrt{4+e^{2}}} $

$ \frac{(1+2 e)}{\sqrt{4+e^{2}}} $

If $\sqrt{1-x^2}+\sqrt{1-y^2}=\sqrt{3}(x-y)$, then $d y / d x$ is equal to

$\sqrt{\left(1-x^2\right)\left(1-y^2\right)}$

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

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

Let $f\left( x \right) = {e^x},{\rm{g}}\left( x \right) = {\sin ^{ - 1}}x\,$ and $h\left( x \right) = f\left[ {{\rm{g}}\left( x \right)} \right],$ then $\frac{{h'\left( x \right)}}{{h\left( x \right)}}$ is equal to

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

If $\sin y = x\sin \left( {a + y} \right),$ then $\frac{{dy}}{{dx}}$ is

$\frac{{\sin a}}{{{{\sin }^2}\left( {a + y} \right)}}$

$\sin a{\sin ^2}\left( {a + y} \right)$

$\frac{{{\rm{si}}{{\rm{n}}^2}\left( {a - {\rm{y}}} \right)}}{{\sin a}}$

The derivative of $y = {x^{{\rm{In}}\,x}}$ is

$2{x^{{\rm{In}}\,\,x - 1}}$ In $x$

${x^{{\rm{In}}\,\,x - 1}}$ In $x$

If $x \log _e\left(\log _e x\right)-x^2+y^2=4(y > 0)$, then $d y / d x$ at ${x}={e}$ is equal to:

$\frac{(1+2 {e})}{2 \sqrt{4+{e}^2}}$

$\frac{(1+2 {e})}{\sqrt{4+{e}^2}}$

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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Mixed difficulty

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Showing 721-750 of 932 questions

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721 Easy

721 of 932

Application +4 | -1

Let

g

be the inverse function of a differentiable function

f

and

G(x)=\frac{1}{g(x)}

. If

f(4)=2

and

f^{\prime}(4)=\frac{1}{16}

, then the value of

\left(G^{\prime}(2)\right)^2

equals to

View answer and explanation

Correct option: A

Option A (1) is correct because it is the square of the derivative of G(x) evaluated at x=2, found using the inverse function derivative.

722 Easy

722 of 932

Application +4 | -1

A value of

x

in the interval (1,2) such that

f'\left( x \right) = 0

, where

View answer and explanation

Correct option: A

\frac{{3 + \sqrt 3 }}{3}

Option A (

\frac{{3 + \sqrt 3 }}{3}

) is correct because it is the only root of the derivative that lies within the specified interval (1, 2).

723 Medium

723 of 932

Application +4 | -1

x \log _{e}\left(\log _{e} x\right)-x^{2}+y^{2}=4(y>0)

, then

\frac{d y}{d x}

\mathrm{x}=\mathrm{e}

is equal to :

View answer and explanation

Correct option: B

\frac{(2 e-1)}{2 \sqrt{4+e^{2}}}

Option B (

\frac{(2 e-1)}{2 \sqrt{4+e^{2}}}

) is correct because it is the result of differentiating the given equation implicitly with respect to x and substituting x=e and y=

\sqrt{4+e^{2}}

724 Easy

724 of 932

Application +4 | -1

f\left( x \right) = \sqrt {ax} + \frac{{{a^2}}}{{\sqrt {ax} }}

, then

f'\left( a \right)

is equal to

View answer and explanation

Correct option: C

Option C (0) is correct because after differentiating the given function and substituting x = a, the expression simplifies to zero.

725 Easy

725 of 932

Analyzing +4 | -1

The number of values of

x

at which

f(x)=\operatorname{Sin}^{-1}\left(\frac{2 x}{1+x^2}\right)

is not differentiable is

View answer and explanation

Correct option: C

Option C (2) is correct because the function is not differentiable at x = 1 and x = -1 due to the absolute value in the derivative.

726 Medium

726 of 932

Application +4 | -1

Let

f(x)

be a differentiable function at

x = a

with

f'(a) = 2

and

f(a) = 4

. Then,

\lim\limits_{x \to a} \frac{{xf(a) - af(x)}}{{x - a}}

equals

View answer and explanation

Correct option: B

4 - 2a

Option B (4 - 2a) is correct because it is the result of applying L'Hopital's rule or using the definition of the derivative to evaluate the limit.

727 Medium

727 of 932

Analyzing +4 | -1

\sqrt{1-x^2}+\sqrt{1-y^2}=\sqrt{3}(x-y)

, then

d y / d x

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because by substituting x = sin(alpha) and y = sin(beta), and differentiating the resulting expression, we arrive at this result.

728 Easy

728 of 932

Application +4 | -1

y=f(x)

is an odd differentiable function defined on

(-\infty, \infty)

such that

f^{\prime}(3)=-2

, then

f^{\prime}(-3)

equals

View answer and explanation

Correct option: C

-2

Option C (-2) is correct because the derivative of an odd function is an even function, so f'(-3) = f'(3) = -2.

729 Medium

729 of 932

Application +4 | -1

Let

f\left( x \right) = {e^x},\mathrm{g}\left( x \right) = {\sin ^{ - 1}}x\,

and

h\left( x \right) = f\left[ {\mathrm{g}\left( x \right)} \right],

then

\frac{{h'\left( x \right)}}{{h\left( x \right)}}

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 finding h(x) and differentiating, the expression

\frac{{h'(x)}}{{h(x)}}

simplifies to

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

730 Easy

730 of 932

Application +4 | -1

Derivative of

{x^6} + {6^x}

with respect to

x

View answer and explanation

Correct option: C

6{x^5} + {6^x}\log 6

Option C (

6{x^5} + {6^x}\log 6

) is correct because it is the result of applying the power rule and the exponential derivative rule.

731 Medium

731 of 932

Analyzing +4 | -1

The number of points at which the function

f(x)=

|x+1|+|x-1|

is not differentiable is

View answer and explanation

Correct option: A

Option A (2) is correct because the function is not differentiable at the two points where the absolute value expressions change sign, namely x = -1 and x = 1.

732 Medium

732 of 932

Application +4 | -1

The derivative of

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

View answer and explanation

Correct option: C

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

Option C,

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

, is correct because it represents the derivative of the given function after applying a trigonometric substitution and simplifying the expression.

733 Medium

733 of 932

Analyzing +4 | -1

Let

f:[2,4] \rightarrow \mathbb{R}

be a differentiable function such that

\left(x \log _e x\right) f^{\prime}(x)+\left(\log _e x\right) f(x)+f(x) \geq 1, x \in[2,4]

with

f(2)=\frac{1}{2}

and

f(4)=\frac{1}{2}

.

Consider the following two statements :

(A)

f(x) \leq 1

, for all

x \in[2,4]

(B)

f(x) \geq \frac{1}{8}

, for all

x \in[2,4]

Then,

View answer and explanation

Correct option: C

Both the statements (A) and (B) are true

Option D (

\frac{{5}}{{2}}

) is correct because after finding the possible values of α using the given conditions, the sum of these values is equal to 5/2.

734 Easy

734 of 932

Application +4 | -1

f(x)=|\cos x-\sin x|

, then

f^{\prime}(\pi / 4)

is equal to

View answer and explanation

Correct option: D

None of these

The correct answer is None of these because the left-hand derivative and right-hand derivative at x=π/4 are not equal, meaning the function is not differentiable at that point.

735 Easy

735 of 932

Analyzing +4 | -1

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

then

\frac{d y}{d x}

x=\frac{\pi}{2}

is equal to

View answer and explanation

Correct option: C

\frac{-8}{\pi^2+4}

Option C is correct because it involves simplifying the expression and then differentiating, evaluating the derivative at the specified point.

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736 Easy

736 of 932

Application +4 | -1

\sin y = x\sin \left( {a + y} \right),

then

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

View answer and explanation

Correct option: B

\frac{{{{\sin }^2}\left( {a + y} \right)}}{{\sin a}}

Option B (

\frac{{{{\sin }^2}\left( {a + y} \right)}}{{\sin a}}

) is correct because it is the result of differentiating the given equation and solving for

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

737 Medium

737 of 932

Application +4 | -1

Derivative of the function

f\left( x \right) = {\log _5}({\log _7}x),x > 7

View answer and explanation

Correct option: A

\frac{1}{{x(\log 5)(\log 7)({{\log }_7}x)}}

Option A is correct because it is the result of applying the chain rule and the change of base formula for logarithms to find the derivative.

738 Easy

738 of 932

Application +4 | -1

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

then

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

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

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

739 Medium

739 of 932

Application +4 | -1

y=|\cos x|+|\sin x|

, then

\frac{d y}{d x}

x=\frac{2 \pi}{3}

View answer and explanation

Correct option: C

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

Option C is correct because it is the result of differentiating the function in the given neighborhood and evaluating at the specified point.

740 Medium

740 of 932

Analyzing +4 | -1

Let

f(x)= \begin{cases}x e^{a x}, & x \leq 0 \\ x+a x^2-x^3, & x>0\end{cases}

where

a

is a positive constant. The interval in which

f^{\prime \prime}(x) \geq 0

, is

View answer and explanation

Correct option: C

\left[\frac{-2}{a}, \frac{a}{3}\right]

Option C ([

\frac{-2}{a}

\frac{a}{3}

]) is correct because it represents the interval where the second derivative of the given piecewise function is greater than or equal to zero.

741 Medium

741 of 932

Application +4 | -1

The derivative of

y = {x^{\mathrm{In}\,x}}

View answer and explanation

Correct option: C

2{x^{\mathrm{In}\,\,x - 1}}

x

Option C (

2{x^{\mathrm{In}\,x - 1}}

x

) is the correct derivative, obtained by logarithmic differentiation.

742 Easy

742 of 932

Analyzing +4 | -1

f(x)=\left|\begin{array}{ccc}\sec \theta & \tan ^2 \theta & 1 \\ \theta \sec x & \tan x & x \\ 1 & \tan x-\tan \theta & 0\end{array}\right|

, then

f^{\prime}(\theta)

View answer and explanation

Correct option: B

-1

Option B (-1) is correct because after differentiating the determinant and substituting x =

\theta

, the expression simplifies to -1.

743 Easy

743 of 932

Application +4 | -1

Let

x=2 \sin \theta-\sin 2 \theta

and

y=2 \cos \theta-\cos 2 \theta

, then

\left(\frac{d^2 y}{d x^2}\right)_{\theta=\pi}=

View answer and explanation

Correct option: A

\frac{3}{8}

Option A (

\frac{3}{8}

) is correct because it is the value of the second derivative of y with respect to x at the specified point.

744 Medium

744 of 932

Application +4 | -1

y=\tan ^{-1}(\operatorname{cosec} x+\cot x)

, then

\frac{d y}{d x}

equals

View answer and explanation

Correct option: A

-\frac{1}{2}

Option A (

-\frac{1}{2}

) is correct because after simplifying the expression inside the arctangent and differentiating, the derivative is found to be

-\frac{1}{2}

745 Easy

745 of 932

Application +4 | -1

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

then

y^{\prime}(-2)

is equal to

View answer and explanation

Correct option: A

\frac{1}{2 \sqrt{3}}

Option A (

\frac{1}{2 \sqrt{3}}

) is correct because it is the value of the derivative of the given function at x = -2, calculated using the chain rule.

746 Easy

746 of 932

Application +4 | -1

The least value of

g(t)=8 t-t^4

[-2,1]

View answer and explanation

Correct option: C

-32

Option C (-32) is correct because it is the minimum value of the function g(t) on the interval [-2,1], found by evaluating the function at the endpoints and critical points.

747 Medium

747 of 932

Application +4 | -1

x \log _e\left(\log _e x\right)-x^2+y^2=4(y > 0)

, then

d y / d x

{x}={e}

is equal to:

View answer and explanation

Correct option: C

\frac{(2 {e}-1)}{2 \sqrt{4+{e}^2}}

Option C ((2e-1)/(2√(4+e²))) is correct because differentiating the given equation implicitly and substituting x=e allows us to solve for dy/dx, yielding the correct expression.

748 Medium

748 of 932

Application +4 | -1

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

then

\left( {1 + {x^2}} \right)\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}}

View answer and explanation

Correct option: A

{n^2}y

Option A (

{n^2}y

) is correct because after differentiating twice and rearranging, the expression equals

{n^2}y

749 Easy

749 of 932

Application +4 | -1

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

and

\mathrm{g}\left( x \right) = f\left( {f\left( x \right)} \right)

, then for

x > 20,\mathrm{g'}\left( x \right)

is equal to

View answer and explanation

Correct option: C

Option C (1) is correct because for

x > 20

g(x)

simplifies to

x - 4

, and its derivative is 1.

750 Medium

750 of 932

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\frac{-x}{\sqrt{1-x^{4}}}

Option C (

\frac{{-x}}{{\sqrt{{1-x^{4}}}}}

) is correct because by substituting

x^2 = \cos 2\theta

, the expression simplifies to

\theta

, and its derivative is

\frac{{-x}}{{\sqrt{{1-x^{4}}}}}

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