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 $f\left( x \right)$ has a derivative at $x = a,$ then $\mathop {\lim }\limits_{x \to a} \frac{{xf\left( a \right) - af\left( x \right)}}{{x - a}}$ is equal to

$af\left( a \right) - f'\left( a \right)$

$f\left( a \right) + f'\left( a \right)$

$af\left( a \right) + f'\left( a \right)$

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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Medium

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241 Medium

241 of 402

Application +4 | -1

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

, then

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

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because after differentiating the given equation and simplifying, the expression evaluates to 0.

242 Medium

242 of 402

Application +4 | -1

The value of

\frac{d}{{dx}}\left[ {{{\tan }^{ - 1}}\left( {\frac{{\sqrt x \left( {3 - x} \right)}}{{1 - 3x}}} \right)} \right]

View answer and explanation

Correct option: D

\frac{3}{{2\left( {1 + x} \right)\sqrt x }}

Option D (

\frac{3}{{2\left( {1 + x} \right)\sqrt x }}

) is correct because after substituting and simplifying, the derivative equals

\frac{3}{{2\left( {1 + x} \right)\sqrt x }}

243 Medium

243 of 402

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}

244 Medium

244 of 402

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).

245 Medium

245 of 402

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.

246 Medium

246 of 402

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.

247 Medium

247 of 402

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}} }}

248 Medium

248 of 402

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

249 Medium

249 of 402

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.

250 Medium

250 of 402

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.

251 Medium

251 of 402

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.

252 Medium

252 of 402

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.

253 Medium

253 of 402

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.

254 Medium

254 of 402

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.

255 Medium

255 of 402

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.

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256 Medium

256 of 402

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.

257 Medium

257 of 402

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.

258 Medium

258 of 402

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.

259 Medium

259 of 402

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.

260 Medium

260 of 402

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}}}

261 Medium

261 of 402

Analyzing +4 | -1

f\left( x \right)

has a derivative at

x = a,

then

\lim\limits_{x \to a} \frac{{xf\left( a \right) - af\left( x \right)}}{{x - a}}

is equal to

View answer and explanation

Correct option: A

f\left( a \right) - af'\left( a \right)

Option A (

f\left( a \right) - af'\left( a \right)

) is correct because by adding and subtracting

af(a)

in the numerator and applying L'Hopital's rule, the limit simplifies to this expression.

262 Medium

262 of 402

Analyzing +4 | -1

Let

f: R \rightarrow R

be a twice differentiable function such that

\lim \limits_{{x \rightarrow \infty}} f(x)=a(a \neq 0)

, then

\lim \limits_{{x \rightarrow \infty}} x^2 f^{\prime \prime}(x)

is equal to

View answer and explanation

Correct option: D

None of these

Option B (

\frac{{1}}{{16}}

) is correct because it represents the maximum value obtained by completing the square and integrating.

263 Medium

263 of 402

Application +4 | -1

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

, then the value of

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

View answer and explanation

Correct option: A

Option A (0) is correct because substituting the given function and its derivatives into the expression results in zero.

264 Medium

264 of 402

Analyzing +4 | -1

Let

K

be the set of all real values of

x

, where the function

f(x) = \sin \left| x \right| - \left| x \right| + 2(x - \pi )\cos \left| x \right|

is not differentiable. Then, the set

K

is equal to

View answer and explanation

Correct option: B

\phi

(an empty set)

Option B (

\phi

) is correct because the function

f(x)

is differentiable for all real values of

x

, meaning there are no points where it's not differentiable.

265 Medium

265 of 402

Analyzing +4 | -1

f\left( x \right) = a\left| {\sin x} \right| + b{e^{\left| x \right|}} + c{\left| x \right|^3}

and if

f\left( x \right)

is differentiable at

x = 0

, then

View answer and explanation

Correct option: B

a = 0,b = 0;c \in R

Option B (

a = 0,b = 0;c \in R

) is correct because for

f(x)

to be differentiable at

x=0

, the coefficients of non-differentiable terms must be zero.

266 Medium

266 of 402

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: D

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

The correct answer is D (

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

) because it is the value of the derivative of y with respect to x at x=1, after simplifying the function and differentiating.

267 Medium

267 of 402

Application +4 | -1

x=\sin ^{-1}\left(t^2-1\right), y=\cos ^{-1}(2 t)

then

\frac{d y}{d x}

t=0

View answer and explanation

Correct option: A

-\sqrt{2}

Option A (

-\sqrt{2}

) is correct because it is the value of dy/dx at t=0, calculated using parametric differentiation.

268 Medium

268 of 402

Application +4 | -1

Let

f

be a differentiable function satisfying

[f(x)]^n=f(n x)

for all

x \in R

.

Then,

f^{\prime}(x) f(n x)

View answer and explanation

Correct option: C

f(x) f^{\prime}(n x)

Option C (

f(x) f^{\prime}(n x)

) is correct because it is derived by differentiating the given equation and manipulating the result.

269 Medium

269 of 402

Application +4 | -1

Let

f(x) = \sin x,\mathrm{g}\left( x \right) = {x^2}

and

\,h\left( x \right) = {\log _e}x

. If

F\left( x \right) = \left( {hog\,\,of} \right)\left( x \right)

, then

F''\left( x \right)

is equal to

View answer and explanation

Correct option: D

- 2\mathrm{cose}{\mathrm{c}^2}x

Option D (

- 2\mathrm{cose}{\mathrm{c}^2}x

) is correct because after finding F(x) and differentiating twice, the result is

- 2\mathrm{cose}{\mathrm{c}^2}x

270 Medium

270 of 402

Application +4 | -1

The derivative of

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

with respect to

x^{2 n}

View answer and explanation

Correct option: A

-\frac{1}{x^n+x^{3 n}}

Option A (

-\frac{1}{x^n+x^{3 n}}

) is correct because it is the result of differentiating

\sin ^{-1}(\frac{1-x^{2 n}}{1+x^{2 n}})

with respect to

x^{2n}

using the chain rule and simplification.

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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