The function $f$ defined by $f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{\sin {x^2}}}{x},x \ne 0}\\ {0,x = 0} \end{array}} \right.$ is

Neither continuous nor derivable at $x = 0$

Continuous but not derivable at $x = 0$

On maximum point of ${\sin ^p}x\,{\cos ^q}x$ is

$x = {\tan ^{ - 1}}\left( {\sqrt {\frac{q}{p}} } \right)$

$x = {\tan ^{ - 1}}\left( {\frac{p}{q}} \right)$

$x = {\tan ^{ - 1}}\left( {\frac{q}{p}} \right)$

$y y^{\prime \prime}-2\left(y^{\prime}\right)^2+1=0$

$y y^{\prime \prime}+2\left(y^{\prime}\right)^2+1=0$

Differential coefficient of $\sqrt {\sec \sqrt x } $ is

$\frac{1}{{4\sqrt x }}\sec \sqrt x \sin \sqrt x $

$\frac{1}{2}\sqrt x \sec \sqrt x \sin \sqrt x $

$\frac{1}{2}\sqrt x {\left( {\sec \sqrt x } \right)^{\frac{3}{2}}}.\sin \sqrt x $

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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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211 Medium

211 of 932

Application +4 | -1

f(x)=\cos \left(x^2-4[x]\right)

for

0 < x < 1

, where

[x]=

G.I.F

., then

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

is equal to

View answer and explanation

Correct option: A

-\sqrt{\frac{\pi}{2}}

Option A (

-\sqrt{\frac{{\pi}}{{2}}}

) is correct because it is the value of the derivative of the function at the given point.

212 Easy

212 of 932

Application +4 | -1

x=a\left[\cos \theta+\log \tan \frac{\theta}{2}\right],

y=a \sin \theta

then

\frac{d y}{d x}=

View answer and explanation

Correct option: C

\tan \theta

Option C (

\tan \theta

) is correct because it is the result of calculating

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

using parametric differentiation.

213 Easy

213 of 932

Application +4 | -1

f\left( x \right) = {\log _a}\left( {{{\log }_a}x} \right)

, then

f'\left( x \right)

View answer and explanation

Correct option: A

\frac{{{{\log }_a}e}}{{x{{\log }_e}x}}

Option A (

\frac{{{{\log }_a}e}}{{x{{\log }_e}x}}

) is correct because it is the correct derivative of the given logarithmic function using the chain rule and change of base formula.

214 Medium

214 of 932

Analyzing +4 | -1

Let

f

and

g

be twice differentiable functions on

\mathrm{R}

such that

f^{\prime \prime}(x)=g^{\prime \prime}(x)+6 x

f^{\prime}(1)=4 g^{\prime}(1)-3=9

f(2)=3 g(2)=12 .

Then which of the following is NOT true?

View answer and explanation

Correct option: D

-1 < x < 2

, then

|f(x)-g(x)| < 8

Option D is NOT true because when x = -1, |f(x) - g(x)| = 10, which is greater than 8, invalidating the statement that |f(x) - g(x)| < 8 for -1 < x < 2.

215 Medium

215 of 932

Analyzing +4 | -1

Let

g(x)=\log f(x)

, where

f(x)

is twice differentiable positive function on

(0, \infty)

such that

f(x+1)=x f(x)

. Then, for

N=1,2,3, \ldots g^{\prime \prime}\left(N+\frac{1}{2}\right)-g^{\prime \prime}\left(\frac{1}{2}\right)=

View answer and explanation

Correct option: A

-4\left\{1+\frac{1}{9}+\frac{1}{25}+\ldots+\frac{1}{(2 N-1)^2}\right\}

Option A (

-4\left\{1+\frac{1}{9}+ \frac{1}{25}+\ldots+\frac{1}{(2 N-1)^2}\right\}

) is correct because it represents the difference in the second derivative of

g(x)

evaluated at

N+\frac{1}{2}

and

\frac{1}{2}

, derived using the given functional equation.

216 Medium

216 of 932

Analyzing +4 | -1

The function

f

defined by

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\frac{{\sin {x^2}}}{x},x \ne 0}\\ {0,x = 0} \end{array}} \right.

View answer and explanation

Correct option: A

Continuous and derivable at

x = 0

Option A (Continuous and derivable at

x = 0

) is correct because the function is both continuous and differentiable at the specified point.

217 Medium

217 of 932

Application +4 | -1

\mathrm{x}=\frac{\mathrm{t}-\mathrm{e}^{-\mathrm{t}^2}}{2 \mathrm{t}^2}

, then

\left(\mathrm{t} \frac{\mathrm{dx}}{\mathrm{dt}}-\mathrm{e}^{-\mathrm{t}^2}\right)=

View answer and explanation

Correct option: A

\frac{1}{2 \mathrm{t}}-2 \mathrm{x}

Option A (

\frac{1}{2 \mathrm{t}}-2 \mathrm{x}

) is correct because it's the result of correctly differentiating x with respect to t and then substituting back the original expression for x.

218 Medium

218 of 932

Analyzing +4 | -1

On maximum point of

{\sin ^p}x\,{\cos ^q}x

View answer and explanation

Correct option: A

x = {\tan ^{ - 1}}\left( {\sqrt {\frac{p}{q}} } \right)

Option A is correct because the maximum point is found by setting the derivative of the function to zero and solving for x.

219 Easy

219 of 932

Application +4 | -1

x^2+y^2=1

, then

View answer and explanation

Correct option: B

y y^{\prime \prime}+\left(y^{\prime}\right)^2+1=0

Option B (y y'' + (y')² + 1 = 0) is correct because it is the result of differentiating the given equation twice and simplifying.

220 Easy

220 of 932

Application +4 | -1

y=2^{1 / \log _{x} 4}

, then

x

is equal to

View answer and explanation

Correct option: C

y^{2}

Option C is correct because it is the result of simplifying the expression using properties of logarithms.

221 Easy

221 of 932

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

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

The correct answer is A (

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

) because it is the result of differentiating the given expression after simplifying it.

222 Easy

222 of 932

Application +4 | -1

Let

f(x)

be a differentiable function such that

f^{\prime}(x)=\sin x+\sin 4 x \cos x

. Then,

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

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

is equal to

View answer and explanation

Correct option: C

-2 \sqrt{2 \pi}

Option C is correct because it correctly applies the chain rule and trigonometric identities to find the derivative and evaluate it at the given point.

223 Easy

223 of 932

Analyzing +4 | -1

y^{1 / m}=\left[x+\sqrt{1+x^2}\right]

, then

\left(1+x^2\right) y_2+x y_1

is equal to

View answer and explanation

Correct option: A

m^2 y

Option A (

m^2 y

) is correct because it is the result of differentiating the given equation twice and simplifying using the given relation.

224 Medium

224 of 932

Understanding +4 | -1

f:R \to R

is an even function which is twice differentiable on

R

and

f''\left( \pi \right) = 1,

then

f''( - \pi )

is equal to

View answer and explanation

Correct option: C

Option C (1) is correct because the second derivative of an even function is also an even function, and therefore

f''(-\pi) = f''(\pi) = 1

225 Easy

225 of 932

Analyzing +4 | -1

y = {\tan ^{ - 1}}\left( {\frac{{\log \left( {\frac{e}{{{x^2}}}} \right)}}{{\log \left( {e{x^2}} \right)}}} \right) + {\tan ^{ - 1}}\left( {\frac{{3 + 2\log x}}{{1 - 6\log x}}} \right)

, then

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

View answer and explanation

Correct option: C

Option C (0) is correct because after simplifying the expression for y, we find that y is a constant, and therefore its second derivative is 0.

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226 Easy

226 of 932

Application +4 | -1

Differential coefficient of

\sqrt {\sec \sqrt x }

View answer and explanation

Correct option: B

\frac{1}{{4\sqrt x }}{\left( {\sec \sqrt x } \right)^{\frac{3}{2}}}.\sin \sqrt x

Option B (

\frac{1}{{4\sqrt x }}{\left( {\sec \sqrt x } \right)^{\frac{3}{2}}}.\sin \sqrt x

) is correct because it correctly applies the chain rule to find the derivative of the given function.

227 Medium

227 of 932

Application +4 | -1

f\left( {x + y + z} \right) = f\left( x \right).f\left( y \right).f\left( z \right)

for all

x,y,z

and

f\left( 2 \right) = 4,f'\left( 0 \right) = 3

, then

f'\left( 2 \right)

equals

View answer and explanation

Correct option: A

Option A (12) is correct because f'(2) = 4f'(0) = 4 * 3 = 12.

228 Medium

228 of 932

Analyzing +4 | -1

Let

f(x)=

Degree of

\left(u^{x^2}+u^2+2 u+3\right)

. Then, at

x=\sqrt{2}, f(x)

View answer and explanation

Correct option: A

continuous but not differentiable

Option A (continuous but not differentiable) is correct because the function is continuous at x=

\sqrt{2}

but the left-hand and right-hand derivatives are not equal at that point.

229 Easy

229 of 932

Application +4 | -1

f

and

g

are differentiable functions such that

g^{\prime}(a)=2

and

g(a)=b

and if

f \circ g

is an identity function, then

f^{\prime}(b)

has the value equal to

View answer and explanation

Correct option: D

1 / 2

Option D (

\frac{1}{2}

) is correct because it is the value of f'(b) calculated using the properties of inverse functions and the chain rule.

230 Hard

230 of 932

Analyzing +4 | -1

f(x)=\left\{\begin{array}{cc}\frac{x\left(3 e^{1 / x}+4\right)}{2-e^{1 / x}}, x \neq 0 \\ 0, \quad x=0\end{array}\right.

then

f(x)

View answer and explanation

Correct option: B

Continuous but not differentiable at

x=0

Option B (Continuous but not differentiable at

x=0

) is correct because the left-hand and right-hand limits of the function at x=0 are equal, but the left-hand and right-hand derivatives are not equal.

231 Medium

231 of 932

Application +4 | -1

The value of the derivative of

|x-1|+|x-3|

x=2

View answer and explanation

Correct option: B

Option B (0) is correct because at x=2, the function f(x) = |x-1| + |x-3| is differentiable, and its derivative is 0.

232 Easy

232 of 932

Application +4 | -1

If the parametric equation of a curve given by

x = {e^t}\;\mathrm{cost},\;y = {e^t}\sin t

then the tangent to the curve at the point

t = \pi /4

makes with axis of

x

the angle

View answer and explanation

Correct option: D

\frac{\pi }{2}

Option D (

\frac{\pi}{2}

) is correct because the tangent at t =

\frac{\pi}{4}

has an infinite slope, indicating a vertical line.

233 Medium

233 of 932

Application +4 | -1

f(x)=x^3+x^4+\log x

and

g

is the inverse of

f

, then

g^{\prime}(2)

is :

View answer and explanation

Correct option: B

\frac{1}{8}

Option B (

\frac{1}{8}

) is correct because it is the value of

g'(2)

found by using the inverse function derivative formula and evaluating at the correct point.

234 Easy

234 of 932

Application +4 | -1

x=\cos \theta, y=\sin ^3 \theta

, then

\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^2+y \frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}

\theta=\pi / 2

View answer and explanation

Correct option: D

-3

Option D (-3) is correct because after finding the first and second derivatives and substituting

\theta = \frac{{\pi}}{{2}}

, the expression evaluates to -3.

235 Easy

235 of 932

Application +4 | -1

Let

f

and

g

be differentiable functions satisfying

g'\left( a \right) = 2,g\left( a \right) = b

and

fog = I

(identify function). Then,

f'\left( b \right)

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}

Option A (1/2) is correct because, given fog = I, the derivative f'(b) can be found using the chain rule and the fact that g'(a) = 2 and g(a) = b.

236 Medium

236 of 932

Application +4 | -1

y = {\log _{{x^2} + 4}}\left( {7{x^2} - 5x + 1} \right)

, then

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

is equal to

View answer and explanation

Correct option: B

\frac{1}{{{{\log }_e}\left( {{x^2} + 4} \right)}}\left\{ {\frac{{14x - 5}}{{7{x^2} - 5x + 1}} - \frac{{2xy}}{{{x^2} + 4}}} \right\}

Option B is correct because it correctly applies the quotient rule and chain rule to differentiate the logarithmic function with a variable base.

237 Easy

237 of 932

Application +4 | -1

y^x-x^y=1

, then the value of

\frac{d y}{d x}

x=1

View answer and explanation

Correct option: A

2(1-\log 2)

Option A (

2(1-\log 2)

) is correct because it is the result of differentiating the given equation and substituting the given values.

238 Medium

238 of 932

Application +4 | -1

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

equals

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it represents the correct application of the chain rule and inverse function differentiation.

239 Easy

239 of 932

Application +4 | -1

f(x)=x^{2}-x+5, x>\frac{1}{2}

, and

\mathrm{g}(\mathrm{x})

is its inverse function, then

\mathrm{g}^{\prime}(7)

equals:

View answer and explanation

Correct option: C

\frac{1}{3}

Option C (

\frac{1}{3}

) is correct because it is the value of the derivative of the inverse function at x=7.

240 Medium

240 of 932

Analyzing +4 | -1

y = \sqrt {x + \sqrt {y + \sqrt {x + \sqrt {y + \ldots \infty } } } }

, then

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

is equal to

View answer and explanation

Correct option: D

None of these

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

\frac{{dy}}{{dx}} = \frac{{{{y^2} - x}}}{{{{y^3} - xy - 1}}}

does not match any of the given options.

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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