If variables $x$ and $y$ are related by the equation $x = \mathop \smallint \limits_0^y \frac{1}{{\sqrt {1 + 9{u^2}} }}du$, then $\frac{{dy}}{{dx}}$ is equal to

$\frac{1}{{\sqrt {1 + 9{y^2}} }}$

If $y = {\cos ^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right)$, then $\frac{{dy}}{{dx}}$ is

$\frac{{ - 2}}{{1 + {x^2}}}$ for all $x$

$\frac{{ - 2}}{{1 + {x^2}}}$ for all $\left| x \right| > 1$

$\frac{2}{{1 + {x^2}}}$ for all $\left| x \right| > 1$

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

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

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

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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Mixed difficulty

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571 Easy

571 of 932

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.

572 Easy

572 of 932

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}

573 Medium

573 of 932

Application +4 | -1

y=1+\frac{x}{\underline{1}}+\frac{x^2}{\underline{2}}+\frac{x^3}{\underline{3}}+\ldots, \frac{d y}{d x}

is equal to

View answer and explanation

Correct option: D

y

Option D (

y

) is correct because the given series is the expansion of

e^x

, and the derivative of

e^x

e^x

, which is represented by

y

in this case.

574 Medium

574 of 932

Application +4 | -1

x^{y} \cdot y^{x}=16

, then

\frac{d y}{d x}

(2,2)

View answer and explanation

Correct option: A

-1

Option A (-1) is correct because by differentiating the given equation implicitly and substituting the point (2,2), we find dy/dx = -1.

575 Medium

575 of 932

Application +4 | -1

If variables

x

and

y

are related by the equation

x = \int \limits_0^y \frac{1}{{\sqrt {1 + 9{u^2}} }}du

, then

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

is equal to

View answer and explanation

Correct option: B

\sqrt {1 + 9{y^2}}

Option B (

\sqrt {1 + 9{y^2}}

) is correct because it is the direct result of differentiating the inverse function and applying the Fundamental Theorem of Calculus.

576 Easy

576 of 932

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).

577 Easy

577 of 932

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.

578 Medium

578 of 932

Analyzing +4 | -1

Let

f(x)=\int_0^{g(x)} \frac{d t}{\sqrt{1+t^2}}

where

g(x)=\int_0^{\cos x}\left(1+\sin t^2\right) d t

. Also,

h(x)=e^{-|x|}

and

f(x)=x^2 \sin \frac{1}{x}

, if

x \neq 0

and

f(0)=0

, then

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

is equal to

View answer and explanation

Correct option: C

h^{\prime}\left(0^*\right)

Option C (

h^{\prime}(0^+)

) is correct because the derivative of f(x) at

\frac{\pi}{2}

is equal to -1, which is also the right-hand derivative of h(x) at 0.

579 Easy

579 of 932

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}

580 Medium

580 of 932

Application +4 | -1

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

, then

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

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the derivative depends on the value of x, specifically whether |x| > 1 or |x| < 1, due to the absolute value that arises during simplification.

581 Medium

581 of 932

Application +4 | -1

f(9)=9

and

f^{\prime}(9)=4

, then

\lim _{x \rightarrow 9} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3}

equals

View answer and explanation

Correct option: B

Option B (4) is correct because applying L'Hopital's rule and using the given values yields 4.

582 Medium

582 of 932

Application +4 | -1

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

equals

View answer and explanation

Correct option: D

-\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3}

The correct answer is D (

-\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3}

) because it represents the second derivative of x with respect to y in terms of derivatives of y with respect to x.

583 Medium

583 of 932

Application +4 | -1

The solution of the differential equation

\left(x y^2+x\right) d x+\left(y x^2+y\right) d y=0

View answer and explanation

Correct option: A

\left({x}^2+1\right)\left({y}^2+1\right)={c}

Option A (

\left({x}^2+1\right)\left({y}^2+1\right)={c}

) is correct because separating variables and integrating leads to the given solution.

584 Easy

584 of 932

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.

585 Hard

585 of 932

Analyzing +4 | -1

Function

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

View answer and explanation

Correct option: B

Except

x = 1

and

x = 2

differentiable everywhere in

R

Option B (Except x = 1 and x = 2 differentiable everywhere in R) is correct because the function is not differentiable at x=1 and x=2 due to the absolute value functions.

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

586 Easy

586 of 932

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.

587 Easy

587 of 932

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.

588 Easy

588 of 932

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)}

589 Easy

589 of 932

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.

590 Easy

590 of 932

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.

591 Easy

591 of 932

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

592 Hard

592 of 932

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 !}+\frac{f^{i v}(1)}{4 !} \ldots+\frac{(-1)^n f^n(1)}{n !}

, is

View answer and explanation

Correct option: D

Option D is correct because it correctly applies the binomial theorem to the given expression.

593 Medium

593 of 932

Application +4 | -1

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {a{x^2} - b,\left| x \right| < 1}\\ {\frac{1}{{\left| x \right|}},\left| x \right| \ge 1} \end{array}} \right.

is differentiable at

x = 1

, then

View answer and explanation

Correct option: B

a = - \frac{1}{2},b = - \frac{3}{2}

Option B (

a = - \frac{{1}}{{2}},b = - \frac{{3}}{{2}}

) is correct because it satisfies the conditions for differentiability at x=1, ensuring both continuity and equal derivatives from both sides.

594 Easy

594 of 932

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.

595 Easy

595 of 932

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.

596 Medium

596 of 932

Analyzing +4 | -1

{e^y} + xy = e

, then the value of

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

for

x = 0

, is

View answer and explanation

Correct option: B

\frac{1}{{{e^2}}}

Option B (

\frac{1}{{{e^2}}}

) is correct because after differentiating the given equation twice and substituting x=0 and y=1, the second derivative is found to be

\frac{1}{{{e^2}}}

597 Easy

597 of 932

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.

598 Hard

598 of 932

Analyzing +4 | -1

y=\left|x-x^2\right|

, then

\frac{d y}{d x}

x=1

View answer and explanation

Correct option: C

Does not exist

Option C (Does not exist) is correct because the left-hand derivative and right-hand derivative at x=1 are not equal, indicating a non-differentiable point.

599 Hard

599 of 932

Analyzing +4 | -1

y = {\log ^n}x,

where

{\log ^n}

means

log\,\,log\,\,log \ldots

(repeated

n

times), then

x\log x\,\,{\log ^2}x\,\,{\log ^3}x \ldots {\log ^{n - 1}}x\,\,{\log ^n}x\,\,\frac{{dy}}{{dx}}

is equal to

View answer and explanation

Correct option: B

{\log ^n}x

The correct answer is B (

{\log ^n}x

) because it is obtained by repeatedly applying the chain rule to the logarithmic function.

600 Easy

600 of 932

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.

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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