If $f\left( x \right) = \sqrt {{x^2} - 2x + 1} $, then

$f'\left( x \right) = 1$ for all $x$

$f'\left( x \right) = - 1$ for all $x \le 1$

$f'\left( x \right) = 1$ for all $x \ge 1$

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 · Mixed difficulty

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661 Easy

661 of 932

Application +4 | -1

Let

f\left( x \right)

be a polynomial function of the second degree. If

f\left( 1 \right) = f\left( { - 1} \right)

and

{a_1},{a_2},{a_3}

are in AP, then

f'\left( {{a_1}} \right),f'\left( {{a_2}} \right),f'\left( {{a_3}} \right)

are in

View answer and explanation

Correct option: A

Option A (AP) is correct because the derivative of a quadratic function is a linear function, and the values of a linear function at points in arithmetic progression are also in arithmetic progression.

662 Medium

662 of 932

Analyzing +4 | -1

Let

S=\left\{t \in R: f(x)=|x-\pi|\left(e^{|x|}-1\right) \sin |x|\right.

is not differentiable at

t

. Then, the set

S

is equal to

View answer and explanation

Correct option: D

\phi

Option D (

\phi

) is correct because the function

f(x)

is differentiable at both

x=0

and

x=\pi

, meaning the set

S

of non-differentiable points is empty.

663 Medium

663 of 932

Analyzing +4 | -1

The set of points where

f(x)=\frac{x}{1+|x|}

is differentiable is

View answer and explanation

Correct option: C

(-\infty, \infty)

Option C is correct because the function is differentiable for all real numbers, as the derivative exists everywhere.

664 Easy

664 of 932

Application +4 | -1

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

then the value of

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

x = \frac{\pi }{6}

View answer and explanation

Correct option: A

- \frac{1}{2}

Option A is correct because simplifying the expression and then differentiating yields -1/2, which is independent of x.

665 Easy

665 of 932

Analyzing +4 | -1

f(x)=2 \operatorname{Sin}^{-1} \sqrt{1-x}+\operatorname{Sin}^{-1}(2 \sqrt{x(1-x)}

for

0 < x < 1/2

then

f'(x)

is equal to

View answer and explanation

Correct option: B

Option B is correct because the derivative of the given function simplifies to 0 for

0 < x < 1/2

666 Easy

666 of 932

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.

667 Medium

667 of 932

Application +4 | -1

The value of

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

x = \frac{\pi }{2}

, where

y

is given by

y = {x^{\sin x}} + \sqrt x

, is

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is the value of dy/dx at x=pi/2, calculated using logarithmic differentiation.

668 Easy

668 of 932

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).

669 Medium

669 of 932

Application +4 | -1

x^{m } y^n=(x+y)^{n+1}

, then

\frac{d y}{d x}=\ldots \ldots

View answer and explanation

Correct option: A

\frac{y}{x}

Option A (

\frac{y}{x}

) is correct because differentiating the given equation using logarithms and simplifying leads to dy/dx = y/x.

670 Easy

670 of 932

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.

671 Medium

671 of 932

Application +4 | -1

\sec \left( {\frac{{{x^2} - {y^2}}}{{{x^2} + {y^2}}}} \right) = {e^a},

then

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

is equal to

View answer and explanation

Correct option: B

\frac{y}{x}

Option B (y/x) is correct because implicit differentiation and simplification leads to dy/dx = y/x.

672 Medium

672 of 932

Analyzing +4 | -1

f\left( x \right) = \sin x

and

g\left( x \right) = {\mathop\mathrm{s}\nolimits} \,gn\sin x

, then

g'\left( 1 \right)

equals

View answer and explanation

Correct option: C

\cos 1

Option C (

\cos 1

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

g(x)

x=1

673 Medium

673 of 932

Analyzing +4 | -1

f\left( x \right) = \sqrt {{x^2} - 2x + 1}

, then

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the derivative is 1 for x > 1 and -1 for x < 1, and it's not defined at x=1.

674 Easy

674 of 932

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.

675 Easy

675 of 932

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.

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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

676 Easy

676 of 932

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.

677 Medium

677 of 932

Analyzing +4 | -1

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

, then

f\left( x \right)

is differentiable on

View answer and explanation

Correct option: B

R - \left\{ { - 1,1} \right\}

Option B (

R - \\{-1, 1\\}

) is correct because the derivative of

f(x)

exists for all real numbers except

x = -1

and

x = 1

, where the function is not differentiable.

678 Medium

678 of 932

Application +4 | -1

y=\tan ^{-1} \sqrt{\frac{1+\cos x}{1-\cos x}}

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: B

-\frac{1}{2}

Option B (

-\frac{1}{2}

) is correct because after simplifying the expression using trigonometric identities, the derivative is found to be -1/2.

679 Easy

679 of 932

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}

680 Medium

680 of 932

Application +4 | -1

x = a\left( {\mathrm{cos\theta } + \mathrm{\theta sin\theta }} \right)

and

y = a(\mathrm{sin\theta } - \theta \cos \theta ),

then

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

is equal to

View answer and explanation

Correct option: B

\mathrm{tan}\,\theta

Option B (

\mathrm{tan}\,\theta

) is correct because it is the result of dividing

\frac{dy}{d\theta}

\frac{dx}{d\theta}

681 Easy

681 of 932

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}

682 Easy

682 of 932

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.

683 Easy

683 of 932

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.

684 Easy

684 of 932

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.

685 Easy

685 of 932

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.

686 Easy

686 of 932

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.

687 Easy

687 of 932

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.

688 Medium

688 of 932

Application +4 | -1

The derivative of

\sin \left( {{x^3}} \right)

w.r.t.

\cos \left( {{x^3}} \right)

View answer and explanation

Correct option: C

- \cot \left( {{x^3}} \right)

Option C is correct because it uses the chain rule to find the derivative of both functions and then divides them.

689 Easy

689 of 932

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)

690 Medium

690 of 932

Application +4 | -1

f(1)=3, f^{\prime}(1)=-\frac{1}{3}

, then the derivative of

\left|x^{11}+f(x)\right|^{-2}

x=1

, is

View answer and explanation

Correct option: D

f^{\prime}(1)

Option D (

f^{\prime}(1)

) is correct because after applying the chain rule and substituting the given values, the derivative simplifies to

f^{\prime}(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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