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

$ \frac{2(x-\sin x)^{3 / 2}}{\sqrt{x}}\left[\frac{3}{2} \cdot \frac{1-\cos x}{1-\sin x}-\frac{1}{2 x}\right] $

$ \frac{2(x-\sin x)^{1 / 2}}{\sqrt{x}}\left[\frac{3}{2} \cdot \frac{1-\cos x}{x-\sin x}-\frac{1}{2 x}\right] $

The derivative of ${a^{\sec x}}{\rm{w}}{\rm{.r}}{\rm{.t}}{\rm{.}}{a^{\tan x}}(a > 0)$ is

$\sin x{\rm{ }}{a^{\sec x - \tan x}}$

$\sec x{\rm{ }}{a^{\sec x - \tan x}}$

$\sin x{\rm{ }}{a^{\tan x - \sec x}}$

If $a{x^2} + 2hxy + b{y^2} = 1$, then $\frac{{{d^2}y}}{{d{x^2}}}$ equals

$\frac{{{h^2} - ab}}{{{{\left( {hx + by} \right)}^2}}}$

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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Mixed difficulty

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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 541-570 of 932 questions

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541 Easy

541 of 932

Analyzing +4 | -1

Let

f(x)

be a second degree polynomial function such that

f(-1)=f(1)

and

\alpha, \beta, \gamma

are in A.P. Then,

f^{\prime}(\alpha), f^{\prime}(\beta), f^{\prime}(\gamma)

are in

View answer and explanation

Correct option: A

Option A (AP) is correct because the derivative of the quadratic function is linear, and since

\alpha, \beta, \gamma

are in AP, their corresponding derivatives will also be in AP.

542 Easy

542 of 932

Application +4 | -1

\sec \left( {\frac{{x - y}}{{x + y}}} \right) = a

, then

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

View answer and explanation

Correct option: A

\frac{y}{x}

Option A (

\frac{y}{x}

) is correct because it is the correct derivative obtained by implicit differentiation of the given equation.

543 Easy

543 of 932

Application +4 | -1

x^m y^n=(x+y)^{m+n}

, then

\frac{d y}{d x}

View answer and explanation

Correct option: B

\frac{y}{x}

Option B (y/x) is correct because it is the derivative obtained after implicit differentiation and simplification.

544 Easy

544 of 932

Analyzing +4 | -1

The derivative of

f(\tan x)

with respect to

g(\sec x)

x=\frac{\pi}{4}

, where

f^{\prime}(1)=2

and

g^{\prime}(\sqrt{2})=4

, is

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because after applying the chain rule and simplifying, the derivative evaluates to

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

x = \frac{\pi}{4}

545 Easy

545 of 932

Analyzing +4 | -1

Let

f

be a differentiable function satisfying

f(x)+f(y)+f(z)+f(x) f(y) f(z)=14

for all

x, y, z \in R

Then,

View answer and explanation

Correct option: B

f^{\prime}(x)=0

for all

x \in R

Option D (All the above) is correct because all three given statements are true based on the given functions and their derivatives.

546 Easy

546 of 932

Application +4 | -1

f(x)=|\log x|

, then for

x \neq 1, f^{\prime}(x)

equals

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the derivative of

f(x) = |\log x|

is a piecewise function that depends on the value of x.

547 Easy

547 of 932

Analyzing +4 | -1

The 2nd derivative of

a{\sin ^3}t

with respect to

a{\cos ^3}t

\,t = \frac{\pi }{4}

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is the value obtained after calculating the second derivative and substituting

t = \frac{\pi}{4}

548 Easy

548 of 932

Application +4 | -1

\frac{d}{d x}\left[\tan ^{-1} \sqrt{\frac{1-\cos x}{1+\cos x}}\right]

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}

Option C is correct because the expression simplifies to x/2, and its derivative is 1/2.

549 Easy

549 of 932

Analyzing +4 | -1

Let

x = {\log _e}t,t > 0

and

y + 1 = {t^2}.

Then,

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

is equal to

View answer and explanation

Correct option: B

- \frac{1}{2}{e^{ - 4x}}

Option B (

- \frac{1}{2}{e^{ - 4x}}

) is correct because it is the result of differentiating the given equations twice and simplifying.

550 Easy

550 of 932

Application +4 | -1

y=\frac{2(x-\sin x)^{3 / 2}}{\sqrt{x}}

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: B

\frac{2(x-\sin x)^{3 / 2}}{\sqrt{x}}\left[\frac{3}{2} \cdot \frac{1-\cos x}{x-\sin x}-\frac{1}{2 x}\right]

Option B is correct because it follows from logarithmic differentiation and simplification of the derivative of the given function.

551 Medium

551 of 932

Analyzing +4 | -1

y(\alpha)=\sqrt{2\left(\frac{\tan \alpha+\cot \alpha}{1+\tan ^{2} \alpha}\right)+\frac{1}{\sin ^{2} \alpha}}, \alpha \in\left(\frac{3 \pi}{4}, \pi\right)

, then

\frac{d y}{d \alpha}

\alpha=\frac{5 \pi}{6}

is:

View answer and explanation

Correct option: A

Option A (4) is correct because it is the value of the derivative of y with respect to alpha, evaluated at

\alpha = \frac{5\pi}{6}

552 Medium

552 of 932

Application +4 | -1

Which of the following is/are true?

I. The derivative of

f(x)=\sin 2 x

2\left(\cos ^2 x-\sin ^2 x\right)

.

II. The derivative of

g(x)=\cot x

-\operatorname{cosec}^2 x

View answer and explanation

Correct option: A

Both I and II are true

Option A is correct because both statements are true. The derivative of sin(2x) is indeed 2(cos²x - sin²x), and the derivative of cot(x) is -cosec²(x).

553 Medium

553 of 932

Application +4 | -1

\frac{d}{{dx}}\left[ {{x^x} + {x^a} + {a^x} + {a^a}} \right] = \ldots ,a

is constant

View answer and explanation

Correct option: B

{x^x}(1 + \log x) + a.{x^{a - 1}} + {a^x}\log a

Option B (

{x^x}(1 + \log x) + a.{x^{a - 1}} + {a^x}\log a

) is correct because it is the result of differentiating each term in the given expression with respect to x.

554 Easy

554 of 932

Application +4 | -1

Let

f(x)=x[x]

where

[x]

denotes the integral part of

x

. If

x

is not an integer, then

f^{\prime}(x)

is equal to

View answer and explanation

Correct option: A

[x]

Option A (

[x]

) is correct because the derivative of

f(x) = x[x]

when

x

is not an integer is simply

[x]

555 Hard

555 of 932

Application +4 | -1

x\sqrt {1 + y} + y\sqrt {1 + x} = 0

, then

\frac{{d\mathrm{y}}}{{\mathrm{d}x}}

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the derivative of the function obtained after simplifying the given equation.

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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

556 Medium

556 of 932

Analyzing +4 | -1

y = \sin px

and

{y_n}

is the

nth

derivative of

y

, then

\left| {\begin{array}{cccccccccccccccccccc} y&{{y_1}}&{{y_2}}\\ {{y_3}}&{{y_4}}&{{y_5}}\\ {{y_6}}&{{y_7}}&{{y_8}} \end{array}} \right|

is equal to

View answer and explanation

Correct option: B

The correct answer is B (0) because the determinant of the matrix is zero due to identical rows after simplification.

557 Easy

557 of 932

Analyzing +4 | -1

The derivative of

{a^{\sec x}}\mathrm{w}\mathrm{.r}\mathrm{.t}\mathrm{.}{a^{\tan x}}(a > 0)

View answer and explanation

Correct option: C

\sin x\mathrm{ }{a^{\sec x - \tan x}}

Option C (

\sin x\mathrm{ }{a^{\sec x - \tan x}}

) is correct because after applying the chain rule and simplifying, the derivative of

{a^{\sec x}}

with respect to

{a^{\tan x}}

\sin x\mathrm{ }{a^{\sec x - \tan x}}

558 Medium

558 of 932

Application +4 | -1

The differential coefficient 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 after substituting

x = \cos 2\theta

and simplifying, the derivative of the resulting expression is

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

559 Easy

559 of 932

Application +4 | -1

x=\sin ^{-1} t

and

y=\log \left(1-t^2\right)

; then

\left.\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}\right|_{t=1 / 2}

View answer and explanation

Correct option: A

-8 / 3

Option A (-8/3) is correct because it is the value of the second derivative of

y

with respect to

x

evaluated at

t = 1/2

560 Medium

560 of 932

Application +4 | -1

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

then the value of

x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}}

View answer and explanation

Correct option: A

Option A (0) is correct because the expression is a homogeneous function of degree 0, and therefore, by Euler's theorem,

x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}} = 0

561 Easy

561 of 932

Application +4 | -1

The function

y=\sin ^{-1} x

satisfies

View answer and explanation

Correct option: B

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

Option B ((1-x²)y'' = xy') is correct because it is the result of differentiating y=sin⁻¹x twice and rearranging the terms.

562 Medium

562 of 932

Analyzing +4 | -1

For the curve

2 y=x^y+y^x

The value of

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

\mathrm{x}=1

\alpha

then

|\alpha|=

View answer and explanation

Correct option: A

-1

Option C (12) is correct because the least value of the sum of squares is found using Cauchy-Schwarz inequality.

563 Easy

563 of 932

Analyzing +4 | -1

f(x)

is a polynomial satisfying

f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3)

for all

x \in R.

Then,

View answer and explanation

Correct option: D

All the above

Option D (All the above) is correct because the polynomial f(x) satisfies all the given conditions when the values are substituted.

564 Easy

564 of 932

Application +4 | -1

y

is a function of

x

and

\log \left( {x + y} \right) = 2xy,

the value of

y'\left( 0 \right)

is equal to

View answer and explanation

Correct option: A

Option A (1) is correct because, after differentiating the given equation and substituting x=0, the value of

y'(0)

is found to be 1.

565 Easy

565 of 932

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.

566 Hard

566 of 932

Analyzing +4 | -1

a{x^2} + 2hxy + b{y^2} = 1

, then

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

equals

View answer and explanation

Correct option: D

\frac{{{h^2} - ab}}{{{{\left( {hx + by} \right)}^3}}}

Option D is correct because the second derivative is found by implicit differentiation and simplification, resulting in

\frac{{{h^2} - ab}}{{{{\left( {hx + by} \right)}^3}}}

567 Easy

567 of 932

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.

568 Easy

568 of 932

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.

569 Easy

569 of 932

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.

570 Medium

570 of 932

Application +4 | -1

y = \cos (\sin {x^2}),

than at

x = \sqrt {\frac{\pi }{2}} ,\frac{{dy}}{{dx}}

is equal to

View answer and explanation

Correct option: D

Option D (0) is correct because when

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

is substituted into the derivative, the

\cos{\frac{\pi}{2}}

term becomes zero, making the entire expression zero.

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