Find $\frac{{dy}}{{dx}},$ if $x = 2{\rm{cos\theta }} - \cos 2{\rm{\theta }}$ and $y = 2\sin {\rm{\theta }} - \sin 2{\rm{\theta }}{\rm{.}}$

$\tan \frac{{3{\rm{\theta }}}}{2}$

$ - \tan \frac{{3{\rm{\theta }}}}{2}$

$\cot \frac{{3{\rm{\theta }}}}{2}$

$ - \cot \frac{{3{\rm{\theta }}}}{2}$

If $y = {\sin ^n}x\cos nx,$ then $\frac{{dy}}{{dx}}$ is

$n{\sin ^{n - 1}}x\sin \left( {n + 1} \right)x$

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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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361 Easy

361 of 500

Application +4 | -1

f: R \rightarrow R

is defined by

f(x) = \left\{ {\begin{array}{cccccccccccccccccccc} {\frac{{x - 2}}{{{x^2} - 3x + 2}},}&{\mathrm{ if \,\,}x \in R - \{ 1,2\} }\\ {2,}&{\mathrm{ if\,\, }x = 1}\\ {1,}&{\mathrm{ if\,\, }x = 2} \end{array}} \right.

then

\lim\limits _{x \rightarrow 2} \frac{f(x)-f(2)}{x-2}=

View answer and explanation

Correct option: B

-1

Option B (-1) is correct because the limit of the difference quotient simplifies to -1 after factoring and canceling terms.

362 Easy

362 of 500

Application +4 | -1

Let

f(x)

be a polynomial function and

g(x)=e^x

. If

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

, then

h^{\prime \prime}(x)

View answer and explanation

Correct option: D

f^{\prime \prime}\left(e^x\right) \cdot e^{2 x}+f^{\prime}\left(e^x\right) \cdot e^x

Option D (

f^{\prime \prime}(e^x) \cdot e^{2 x}+f^{\prime}(e^x) \cdot e^x

) is correct because it is the result of applying the chain rule twice to find the second derivative of the composite function h(x).

363 Easy

363 of 500

Application +4 | -1

y = {x^2}{e^{mx}},

where

m

is a constant,then

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

is equal to

View answer and explanation

Correct option: A

m{e^{mx}}\left( {{m^2}{x^2} + 6mx + 6} \right)

Option A (

m{e^{mx}}\left( {{m^2}{x^2} + 6mx + 6} \right)

) is correct because it is the result of taking the third derivative of the given function using product rule and chain rule.

364 Easy

364 of 500

Analyzing +4 | -1

The derivative of

\sin ^{-1}\left(3 x-4 x^3\right)

with respect to

\sin ^{-1} x

, is

View answer and explanation

Correct option: B

3, for

|x| < \frac{1}{2}

and

-3

for

\frac{1}{2} < |x| < 1

Option B (3, for

|x| < \frac{1}{2}

and

-3

for

\frac{1}{2} < |x| < 1

) is correct because the derivative changes sign depending on the interval of

x

due to the piecewise definition of the inverse sine function.

365 Easy

365 of 500

Application +4 | -1

x=3 \tan t

and

y=3 \sec t

, then the value of

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

t=\frac{\pi}{4}

, is

View answer and explanation

Correct option: D

\frac{1}{6 \sqrt{2}}

Option D (

\frac{1}{6 \sqrt{2}}

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

366 Easy

366 of 500

Application +4 | -1

\frac{d}{d x}\left[e^{x} \log \left(1+x^{2}\right)\right]

is equal to

View answer and explanation

Correct option: A

e^{x}\left[\log \left(1+x^{2}\right)+\frac{2 x}{1+x^{2}}\right]

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

367 Easy

367 of 500

Application +4 | -1

y = a{\sin ^3}\theta

and

x = a{\cos ^3}\theta ,

then at

\theta = \frac{\pi }{3},\frac{{dy}}{{dx}}

is equal to

View answer and explanation

Correct option: B

- \sqrt 3

Option B (

\ - \sqrt 3

) is correct because, after finding

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

and substituting

\theta = \frac{\pi }{3}

, the result is

-\sqrt{3}

368 Easy

368 of 500

Application +4 | -1

Find

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

x = 2\mathrm{cos\theta } - \cos 2\mathrm{\theta }

and

y = 2\sin \mathrm{\theta } - \sin 2\mathrm{\theta }\mathrm{.}

View answer and explanation

Correct option: A

\tan \frac{{3\mathrm{\theta }}}{2}

Option A (

\tan \frac{{3\theta }}{2}

) is correct because after differentiating x and y with respect to

\theta

and simplifying,

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

equals

\tan \frac{{3\theta }}{2}

369 Easy

369 of 500

Application +4 | -1

y=e^{\left(1+\log _{\mathrm{e}} x\right)}

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: A

\mathrm{e}

Option A (e) is correct because after simplifying the expression for y, the derivative dy/dx is simply e.

370 Easy

370 of 500

Application +4 | -1

\sqrt{(x+y)}+\sqrt{(y-x)}=a

, then

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

equals

View answer and explanation

Correct option: C

2 / a^2

Option C (

2 / a^2

) is correct because it is the second derivative of y with respect to x, derived from the given equation.

371 Easy

371 of 500

Application +4 | -1

y = {\sin ^n}x\cos nx,

then

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

View answer and explanation

Correct option: D

n{\sin ^{n - 1}}x\cos \left( {n + 1} \right)x

The correct answer is

n{\sin ^{n - 1}}x\cos \left( {n + 1} \right)x

because it results from applying the product rule and trigonometric identities to the given function.

372 Easy

372 of 500

Understanding +4 | -1

f\left( x \right) = \sum\nolimits_{n = 0}^\infty {\frac{{{x^n}}}{{n!}}{{\left( {{{\log }_e}a} \right)}^n}} ,a > 0,a \ne 0

, then at

x = 0,f\left( x \right)

View answer and explanation

Correct option: B

Everywhere differentiable

The correct answer is B (Everywhere differentiable) because the function simplifies to

a^x

, which is differentiable everywhere.

373 Easy

373 of 500

Application +4 | -1

x=3 \cos t, y=4 \sin t

then

\frac{d y}{d x}

(3 / \sqrt{2}, 2 \sqrt{2})

is equal to

View answer and explanation

Correct option: D

-\frac{4}{3}

Option D (-4/3) is correct because after finding dy/dx and substituting the given point, the result is -4/3.

374 Easy

374 of 500

Application +4 | -1

\frac{d}{{dx}}[\cos {x^0}]

is equal to

View answer and explanation

Correct option: A

- \frac{\pi }{{180}}\sin {x^0}

Option A (

- \frac{\pi }{{180}}\sin {x^0}

) is correct because it involves converting degrees to radians before differentiating and applying the chain rule.

375 Easy

375 of 500

Application +4 | -1

f

and

g

are differentiable function in

[0,1]

satisfying

f(0)=2=g(1), g(0)=0

and

f(1)=6

, then for some

c \in(0,1):

View answer and explanation

Correct option: B

f^{\prime}(c)=2 g^{\prime}(c)

Option B (

f^{\prime}(c)=2 g^{\prime}(c)

) is correct because it follows from applying Rolle's theorem to the function

h(x) = f(x) - 2g(x)

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376 Easy

376 of 500

Analyzing +4 | -1

f\left( {x,y} \right) = 2{\left( {x - y} \right)^2} - {x^4} - {y^4},

then

{\left| {\left( {{f_{xx}}{f_{yy}} - f_{xy}^2} \right)} \right|_{0,0}}

View answer and explanation

Correct option: C

Option C (0) is correct because after calculating the second-order partial derivatives and substituting (0,0), the expression evaluates to 0.

377 Easy

377 of 500

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it results from simplifying the expression using trigonometric identities and then differentiating.

378 Easy

378 of 500

Analyzing +4 | -1

y = \frac{{(a - x)\sqrt {a - x} - (b - x)\sqrt {x - b} }}{{\sqrt {a - x} + \sqrt {x - b} }}

, then find

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

whenever defined.

View answer and explanation

Correct option: C

\frac{{2x - (a + b)}}{{2\sqrt {(a - x)(x - b)} }}

Option C (

\frac{{2x - (a + b)}}{{2\sqrt {(a - x)(x - b)} }}

) is correct because it is the derivative of the given function after simplification.

379 Easy

379 of 500

Application +4 | -1

Derivative of

{\log _{10}}x

with respect to

{x^2}

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the result of differentiating log base 10 of x with respect to x squared using the chain rule.

380 Easy

380 of 500

Application +4 | -1

The value of

\frac{d}{d x}(|x-1|+|x-5|)

x=3

View answer and explanation

Correct option: B

Option B (0) is correct because at

x=3

, the function

f(x) = |x-1| + |x-5|

is constant, hence its derivative is zero.

381 Easy

381 of 500

Analyzing +4 | -1

Let

f\left( x \right) = {\left( {{x^3} + 2} \right)^{30}}

. If

{f^n}\left( x \right)

is a polynomial of degree 20, where

{f^n}\left( x \right)

denotes the

{n^{th}}

order derivative of

f\left( x \right)

with respect to

x

, then the value of

n

View answer and explanation

Correct option: C

Option C (70) is correct because the degree of the nth derivative is 90-n, and we want this to be 20, so n=70.

382 Easy

382 of 500

Application +4 | -1

The left-hand derivative of

f(x)=[x] \sin \pi x

x=k

k

being an integer, is

View answer and explanation

Correct option: A

(-1)^k(k-1) \pi

Option A (

(-1)^k(k-1) \pi

) is correct because the left-hand derivative is computed using the limit definition and properties of the sine function near integer values.

383 Easy

383 of 500

Application +4 | -1

y = a\cos (\log x) + b\sin (\log x)

where

a,b

are parameters, then

{x^2}y'' + xy'

is equal to

View answer and explanation

Correct option: B

- y

Option B (

-y

) is correct because after differentiating twice and substituting, the expression

{x^2}y'' + xy'

simplifies to

-y

384 Easy

384 of 500

Application +4 | -1

y=\log x \cdot e^{\left(\tan x+x^{2}\right)}

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: C

e^{\left(\tan x+x^{2}\right)}\left[\frac{1}{x}+\left(\sec ^{2} x+2 x\right) \log x\right]

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

385 Easy

385 of 500

Application +4 | -1

u = {x^2} + {y^2}

and

x = s + 3t,y = 2s - t,

then

\frac{{{d^2}u}}{{d{s^2}}}

is equal to

View answer and explanation

Correct option: D

Option D (10) is correct because it is the result of applying the chain rule and differentiating twice with respect to s.

386 Easy

386 of 500

Application +4 | -1

y=\log _{e^x}(x-2)^2

for

x \neq 0,2

, then

y^{\prime}(3)

is equal to

View answer and explanation

Correct option: B

2 / 3

Option B (2/3) is correct because after applying the logarithmic and derivative rules, substituting x=3 gives 2/3.

387 Easy

387 of 500

Analyzing +4 | -1

f\left( x \right) = \cos x.\cos 2x.\cos 4x.\cos 8x.\cos 16x,

then the value of

f'\left( {\frac{\pi }{4}} \right)

View answer and explanation

Correct option: B

\sqrt 2

Option B (

\sqrt 2

) is correct because after simplifying the function using trigonometric identities and then differentiating, the value of f'(

\frac{\pi}{4}

) is

\sqrt 2

388 Easy

388 of 500

Application +4 | -1

Let

(x, y)

be any point on the unit circle with centre at the origin, then

View answer and explanation

Correct option: B

y y^{\prime \prime}+y^{\prime 2}+1=0

Option B (

y y^{\prime \prime}+y^{\prime 2}+1=0

) is correct because it is the result of differentiating the equation of the unit circle twice.

389 Easy

389 of 500

Application +4 | -1

\frac{d}{{dx}}{\left( {\log x} \right)^4}

is equal to

View answer and explanation

Correct option: C

\frac{{4{{\left( {\log x} \right)}^3}}}{x}

Option C (

\frac{{4{{\left( {\log x} \right)}^3}}}{x}

) is correct because it is the result of applying the chain rule to differentiate (log x)⁴.

390 Easy

390 of 500

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.

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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