If $ x=a(t-\sin t) $ and $ y=a(1-\cos t) $, then $ \frac{d y}{d x} $ is equal to

$ \tan \left(\frac{t}{2}\right) $

$ -\tan \left(\frac{t}{2}\right) $

If for $ x \in\left(0, \frac{1}{4}\right) $, the derivative of $ \tan ^{-1}\left(\frac{6 x \sqrt{x}}{1-9 x^{3}}\right) $ is $ \sqrt{\mathrm{x}} \cdot \mathrm{g}(\mathrm{x}) $, then $ g(x) $ equals :

$ \frac{3 x \sqrt{x}}{1-9 x^{3}} $

$ \frac{3 \mathrm{x}}{1-9 \mathrm{x}^{3}} $

If $ y=\frac{e^{x} \log x}{x^{2}} $, then $ \frac{d y}{d x} $ is equal to

$ \frac{e^{x}[1-(x-2) \log x]}{x^{4}} $

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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Easy

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Practice format and chapter quiz

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 481-500 of 500 questions

481 Easy

481 of 500

Application +4 | -1

y=(1+x)\left(1+x^2\right)\left(1+x^4\right) \ldots \left(1+x^{2^n}\right)

then

\frac{d y}{d x}

x=0

is-

View answer and explanation

Correct option: A

Option A (1) is correct because after multiplying by (1-x) and simplifying, the derivative at x=0 is found to be 1.

482 Easy

482 of 500

Analyzing +4 | -1

Let

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

where

[x]

denotes the integer part of

x

. Then at

x=k, k

being an integer, the left derivative of

f(x)

View answer and explanation

Correct option: A

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

Option A is correct because the left derivative of

f(x)

x=k

simplifies to

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

483 Easy

483 of 500

Application +4 | -1

f(4)=4, f^{\prime}(4)=1

, then

\lim\limits _{x \rightarrow 4} \frac{2-\sqrt{f(x)}}{2-\sqrt{x}}

is equal to

View answer and explanation

Correct option: B

Option B (1) is correct because after rationalizing and applying L'Hopital's rule or using the definition of the derivative, the limit evaluates to 1.

484 Easy

484 of 500

Application +4 | -1

x=a(t-\sin t)

and

y=a(1-\cos t)

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: C

\cot \left(\frac{t}{2}\right)

Option C (

\cot \left(\frac{t}{2}\right)

) is correct because it is the simplified form of

\frac{dy}{dx}

after applying trigonometric identities.

485 Easy

485 of 500

Application +4 | -1

The length

l

of a rectangle is decreasing at the rate of

2 \mathrm{~cm} / \mathrm{sec}

while the widths

w

is increasing at the rate of

2 \mathrm{~cm} / \mathrm{sec}

. When

l=12

and

w=5

, the rate of change of area is (in

\mathrm{cm}^2 / \mathrm{sec}

)

View answer and explanation

Correct option: A

Option A (14) is correct because it is the rate of change of the area calculated using the product rule of differentiation.

486 Easy

486 of 500

Analyzing +4 | -1

Let

\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}

be a function such that

|f(x)| \leq x^{2}

, for all

x \in R

. Then, at

x=0

\mathrm{f}

is:

View answer and explanation

Correct option: B

continuous as well as differentiable.

Option B is correct because the given condition |f(x)| ≤ x² implies that f(0) = 0 and f'(0) = 0, making the function both continuous and differentiable at x=0.

487 Easy

487 of 500

Application +4 | -1

{x^p}{y^q} = {\left( {x + y} \right)^{p + q}}

, then

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

is equal to

View answer and explanation

Correct option: A

\frac{y}{x}

Option A,

\frac{y}{x}

, is correct because it is the simplified result of differentiating the given equation

{x^p}{y^q} = {\left( {x + y} \right)^{p + q}}

implicitly with respect to x.

488 Easy

488 of 500

Application +4 | -1

If for

x \in\left(0, \frac{1}{4}\right)

, the derivative of

\tan ^{-1}\left(\frac{6 x \sqrt{x}}{1-9 x^{3}}\right)

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

, then

g(x)

equals :

View answer and explanation

Correct option: B

\frac{9}{1+9 x^{3}}

Option B (

\frac{9}{1+9 x^{3}}

) is correct because it is the expression for g(x) after simplifying and differentiating the given function.

489 Easy

489 of 500

Application +4 | -1

f^{\prime \prime}(x)=-f(x)

, where

f(x)

is a continuous double differentiable function and

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

.

If

F(x)=\left\{f\left(\frac{x}{2}\right)\right\}^2+\left\{g\left(\frac{x}{2}\right)\right\}^2

and

F(5)=5

, then

F(10)

View answer and explanation

Correct option: B

Option B (5) is correct because F(x) is a constant function, and since F(5) = 5, then F(10) = 5.

490 Easy

490 of 500

Application +4 | -1

The derivative of

\left(x^2+1\right) \cos x

is given by

View answer and explanation

Correct option: A

-x^2 \sin x-\sin x+2 x \cos x

Option A (

-x^2 \sin x - \sin x + 2x \cos x

) is correct because it is the result of applying the product rule to differentiate the given function.

491 Easy

491 of 500

Understanding +4 | -1

f:R \to R

is an even function having derivatives of all orders, then an odd function among the following is

View answer and explanation

Correct option: B

f'''

Option B (

f'''

) is correct because the third derivative of an even function is an odd function.

492 Easy

492 of 500

Analyzing +4 | -1

For

x \in R, f(x)=|\log 2-\sin x|

and

g(x)=f(f(x))

, the

View answer and explanation

Correct option: B

g^{\prime}(0)=\cos (\log 2)

Option B (

g^{\prime}(0)=\cos (\log 2)

) is correct because after analyzing the function

g(x)

near

x=0

and applying the chain rule, the derivative at

x=0

simplifies to

\cos(\log 2)

493 Easy

493 of 500

Application +4 | -1

y

is a function of

x

and

\log (x+y)=2 x y

, then the value of

y^{\prime}(0)

View answer and explanation

Correct option: A

Option A (1) is correct because by differentiating the given equation and substituting x=0 and y=1, we find y'(0) = 1.

494 Easy

494 of 500

Application +4 | -1

y=\frac{e^{x} \log x}{x^{2}}

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: D

\frac{e^{x}[1+(x-2) \log x]}{x^{3}}

Option D is correct because the derivative is found using the product and quotient rules, simplifying to the given expression.

495 Easy

495 of 500

Understanding +4 | -1

y=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots \infty

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: A

y

Option A (

y

) is correct because the derivative of the given series, which represents

e^x

, is equal to the original series.

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496 Easy

496 of 500

Application +4 | -1

A point on the curve

y=x^3-3 x+5

at which the tangent line is parallel to

y=-2 x

View answer and explanation

Correct option: C

\left(\frac{1}{\sqrt{3}}, 5-\frac{8 \sqrt{3}}{9}\right)

Option C

\left(\frac{{1}}{{\sqrt{{3}}}}, 5-\frac{{8 \sqrt{{3}}}}{{9}}\right)

is correct because at this point, the derivative of the curve equals the slope of the given line.

497 Easy

497 of 500

Application +4 | -1

The derivative of

{e^x}^3

with respect to

\log x

View answer and explanation

Correct option: C

3{x^3}{e^x}^3

Option C (

3{x^3}{e^x}^3

) is correct because it is the result of correctly applying the chain rule and dividing the derivatives.

498 Easy

498 of 500

Application +4 | -1

y=x \log x-x

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: C

\log x

Option C (

\log x

) is correct because the derivative of

x \log x - x

with respect to

x

\log x

499 Easy

499 of 500

Analyzing +4 | -1

Let

u(x)

and

v(x)

be differentiable functions such that

\frac{u(x)}{v(x)}=7

. If

\frac{u^{\prime}(x)}{v^{\prime}(x)}=p

and

\left(\frac{u(x)}{v(x)}\right)^{\prime}=q

, then

\frac{p+q}{p-q}

has the value equal to

View answer and explanation

Correct option: A

Option A (1) is correct because given the conditions, the value of

\frac{p+q}{p-q}

simplifies to 1.

500 Easy

500 of 500

Application +4 | -1

5 f(x)+3 f\left(\frac{1}{x}\right)=x+2

and

y=x f(x)

, then

\frac{d y}{d x}

x=1

is equal to

View answer and explanation

Correct option: B

\frac{7}{8}

Option B is correct because it correctly solves the system of equations to find f(x) and then calculates the derivative and evaluates it at x=1.

Revision notes and key takeaways

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