If $y=\log _{\cos x} \sin x$, then $\frac{d y}{d x}$ is equal to

$(\tan x \log \cos x+\cot x \log \sin x) /(\log \cos x)^2$

$(\cot x \log \cos x+\tan x \log \sin x) /(\log \sin x)^2$

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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 421-450 of 500 questions

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421 Easy

421 of 500

Analyzing +4 | -1

y=\left[x+\sqrt{x^{2}-1}\right]^{15}+\left[x-\sqrt{x^{2}-1}\right]^{15}

, then

\left(\mathrm{x}^{2}-1\right) \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}

is equal to

View answer and explanation

Correct option: B

224 y^{2}

Option D (

225 y

) is correct. By differentiating the given equation twice and substituting, we find that

(x^{2}-1) \frac{d^{2} y}{dx^{2}}+x \frac{dy}{dx} = 225y

422 Easy

422 of 500

Analyzing +4 | -1

f(x)=\sqrt{x+2 \sqrt{2 x-4}}+\sqrt{x-2 \sqrt{2 x-4}}

, then the value of

10 f^{\prime}\left(102^{+}\right)

, is

View answer and explanation

Correct option: C

Option C (1) is correct because after simplifying the function and finding its derivative, the value of

10f'(102^+)

is 1.

423 Easy

423 of 500

Application +4 | -1

f(x)

is a differentiable function such that

f^{\prime}(a)=2, f(a)=4

, then the value of

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

View answer and explanation

Correct option: B

2 a-4

Option B (

2a-4

) is correct because it is the result of applying L'Hopital's rule or algebraic manipulation to evaluate the limit using the given derivative and function value.

424 Easy

424 of 500

Analyzing +4 | -1

Let

f(x)=x|x|

and

g(x)=\sin x

.
Statement-1 : gof is differentiable at

x=0

and its derivative is continuous at that point.
Statement-2 : gof is twice differentiable at

x=0

. [2009]

View answer and explanation

Correct option: B

Statement-1 is true, Statement-2 is false.

Option B is correct. gof is differentiable at x=0 and its derivative is continuous at that point, but gof is not twice differentiable at x=0.

425 Easy

425 of 500

Application +4 | -1

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

, then

\left( {2y - 1} \right)\frac{{dy}}{{dx}}

is equal to

View answer and explanation

Correct option: C

\cos x

Option C (

\cos x

) is correct because it is the result of differentiating the given equation and solving for (2y-1)dy/dx.

426 Easy

426 of 500

Understanding +4 | -1

f(x)= \begin{cases}x \sin (1 / x), & \text { for } x \neq 0 \\ 0, & \text { for } x=0\end{cases}

then

View answer and explanation

Correct option: A

f

is a continuous function

Option A (

f

is a continuous function) is correct because

f(x)

approaches 0 as

x

approaches 0, which matches the defined value of

f(0) = 0

427 Easy

427 of 500

Application +4 | -1

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

, then

y'\left( 0 \right)

View answer and explanation

Correct option: A

\frac{1}{2}

Option A (

\frac{1}{2}

) is correct because after simplification using trigonometric identities and substitution, the derivative evaluates to 1/2 at x=0.

428 Easy

428 of 500

Application +4 | -1

f

is a bijection satisfying

f^{\prime}(x)=\sqrt{1-\{f(x)\}^2}

, then

\left(f^{-1}\right)^{\prime}(x)

View answer and explanation

Correct option: A

Is equal to

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

Option A is correct because it correctly applies the derivative rules for inverse functions and uses the given condition f'(x).

429 Easy

429 of 500

Application +4 | -1

x\sqrt {1 + y} + y\sqrt {1 + x} = 0

, then

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

View answer and explanation

Correct option: B

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

Option B (

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

is correct because after simplifying the given equation and differentiating, the result is

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

430 Easy

430 of 500

Application +4 | -1

2 x^3-3 x^2 y^2+4 x-y+7=0

and

y(1)=1

, then the value of

y^n(1)

is equal to

View answer and explanation

Correct option: B

-\frac{474}{343}

Option B (

-\frac{474}{343}

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

y''(1)

obtained by implicit differentiation of the given equation and using the initial condition

y(1)=1

431 Easy

431 of 500

Application +4 | -1

f\left( x \right) = {\left| x \right|^3},

then

f'\left( 0 \right)

equal to

View answer and explanation

Correct option: A

Option A (0) is correct because the derivative from the left and right at x=0 both equal 0.

432 Easy

432 of 500

Application +4 | -1

f(x)=\cos \left\{\frac{\pi}{2}[x]-x^3\right\}, 1 < x < 2

, and

[x]

denotes the greatest integer less than or equal to

x

, then the value of

f^{\prime}\left(\sqrt[3]{\frac{\pi}{2}}\right)

, is

View answer and explanation

Correct option: A

The correct answer is A (0). The derivative is found by first simplifying the function using the greatest integer function and then differentiating.

433 Easy

433 of 500

Application +4 | -1

If the angle made by the tangent at the point

({x_0},\,\,{y_0})

on the curve

x = 12(t + \sin t\,\cos t),

y = 12{(1 + \sin t)^2},\,\,\,\,\,0 < t < \frac{\pi }{2},

with the positive x-axis is

\frac{\pi }{3}

then

{y_0}

is equal to:

View answer and explanation

Correct option: C

Option A (60) is correct because it is the value of 25M, where M is the mean deviation of the given numbers about the mean, calculated using the given mean and variance.

434 Easy

434 of 500

Understanding +4 | -1

f: \mathbb{R} \rightarrow \mathbb{R}

is an even function, then

View answer and explanation

Correct option: A

f^{\prime}(0)=0

Option A (

f^{\prime}(0)=0

) is correct because the derivative of an even function is an odd function, and odd functions are zero at x=0.

435 Easy

435 of 500

Application +4 | -1

Which of the following is/are true?

I. The derivative of

y=2 x-\frac{3}{4}

2.

II. The derivative of

y=\left(5 x^3+3 x-1\right)(x-1)

20 x^3+15 x^2+6 x-4

View answer and explanation

Correct option: B

Only I is true

Option B is correct because only statement I is true. The derivative of y=2x - 3/4 is 2, while the derivative of y=(5x³+3x-1)(x-1) is 20x³ - 15x² + 6x - 4, not 20x³+15x²+6x-4.

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436 Easy

436 of 500

Analyzing +4 | -1

f(x)=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right), x \in R

, then

f^{\prime}(x)

is given by

View answer and explanation

Correct option: A

f^{\prime}(x)=\frac{2}{1+x^2}

for all

x \in R-\{-1,1\}

Option A is correct because the derivative of arctan(2x/(1-x²)) is 2/(1+x²) for all x in R - {-1,1}, where the function is not continuous.

437 Easy

437 of 500

Application +4 | -1

y=\sin x-\cos x

and if

f(x)=\frac{d^{17} y}{d x^{17}}

, then the value of

\left(\frac{1024}{\sqrt{2}}\right) f\left(\frac{\pi}{4}\right)

must be:

View answer and explanation

Correct option: A

1024

Option A (1024) is correct because after finding the 17th derivative of

y

and evaluating it at

\frac{\pi}{4}

, we multiply by

\frac{1024}{\sqrt{2}}

to get 1024.

438 Easy

438 of 500

Application +4 | -1

y=a^{x} \cdot b^{2 x-1}

, then

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

View answer and explanation

Correct option: D

y \cdot\left(\log a b^{2}\right)^{2}

Option D (

y \cdot\left(\log a b^{2}\right)^{2}

) is correct because the second derivative of

y

involves multiplying by

(\log a + 2 \log b)

twice, which is equivalent to

y \cdot (\log ab^2)^2

439 Easy

439 of 500

Analyzing +4 | -1

Let

f(x)=x|x|, g(x)=\sin x

and

h(x)=(g o f)(x)

. Then

View answer and explanation

Correct option: C

\mathrm{h}^{\prime}(\mathrm{x})

is continuous at

\mathrm{x}=0

but it is not differentiable at

x=0

Option C is correct. h'(x) is continuous at x=0 but not differentiable because h''(x) has different left and right hand limits at x=0.

440 Easy

440 of 500

Application +4 | -1

xe^{x y}=y+\sin ^2 x

, then at

x=0, \frac{d y}{d x}=

View answer and explanation

Correct option: C

Option C (1) is correct because after differentiating the given equation implicitly and substituting x=0, the value of dy/dx is 1.

441 Easy

441 of 500

Application +4 | -1

\frac{d}{d x}\left(e^{x} \log \sin 2 x\right)

View answer and explanation

Correct option: A

e^{x}(\log \sin 2 x+2 \cot 2 x)

The correct answer is A (

e^{x}(\log \sin 2 x+2 \cot 2 x)

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

442 Easy

442 of 500

Application +4 | -1

y=\log _{\cos x} \sin x

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: A

(\cot x \log \cos x+\tan x \log \sin x) /(\log \cos x)^2

Option A is correct because it is the result of applying the quotient rule and chain rule to differentiate the given logarithmic function.

443 Easy

443 of 500

Application +4 | -1

f(x)=(1+x)^n

, then the value of

f(0)+f^{\prime}(0)+\frac{f^{\prime \prime}(0)}{2 !}+\ldots+\frac{f^n(0)}{n !}

View answer and explanation

Correct option: B

2^n

Option B (

2^n

) is correct because the given expression is the binomial expansion of

(1+1)^n

evaluated at

x=0

444 Easy

444 of 500

Understanding +4 | -1

For a differentiable function

f

\mathbf{R}

there is an

x_0

with

f^{\prime}\left(x_0\right)=\frac{1}{2}\left[f^{\prime}(0)+f^{\prime}(1)\right]

View answer and explanation

Correct option: D

f

is continuously differentiable

Option D (if

f

is continuously differentiable) is correct because the Intermediate Value Theorem can be applied to the derivative of a continuously differentiable function.

445 Easy

445 of 500

Application +4 | -1

f(x)=x+\tan x

and

g(x)

is the inverse of

f(x)

then

g^{\prime}(x)

is equal to

View answer and explanation

Correct option: C

\frac{1}{2+[g(x)-x]^2}

Option B (

\frac{{1}}{{2+[g(x)-x]^2}}

) is correct because by differentiating

f(g(x)) = x

and using trigonometric identities, we arrive at the given expression.

446 Easy

446 of 500

Application +4 | -1

The point of inflection of

y=x^3-5 x^2+

3 x-5

View answer and explanation

Correct option: D

\frac{5}{4}

Option D

\frac{{5}}{{4}}

is correct because it is the x-coordinate where the second derivative of the function is zero and the third derivative is non-zero.

447 Easy

447 of 500

Application +4 | -1

Let

f(x+y)=f(x) f(y)

for all

x

and

y

, and

f(5)=-2, f^{\prime}(0)=3

, then

f^{\prime}(5)

is equal to :

View answer and explanation

Correct option: C

-6

Option C (-6) is correct because it is the value of

f'(5)

found using the given functional equation and derivative at 0.

448 Easy

448 of 500

Application +4 | -1

If the equation

2 \sin ^2 \mathrm{x}+\frac{\sin 2 \mathrm{x}}{2}=\mathrm{k}

has at-least one real solution, then the sum of all integral values of

\mathrm{k}

View answer and explanation

Correct option: A

Option A (3) is correct because the equation can be rewritten as a trigonometric function with a range that allows for only three integer values of k.

449 Easy

449 of 500

Understanding +4 | -1

f:R \to R

is an even function which us twice differentiable on

R

and

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

, then

f''\left( { - \pi } \right)

View answer and explanation

Correct option: C

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

f''(-x) = f''(x)

450 Easy

450 of 500

Application +4 | -1

Let

f(x)=\frac{x^2-x}{x^2+2 x}, x \neq 0,-2

. Then

\frac{d}{d x}\left(f^{-1}(x)\right)

(wherever defined) is equal to

View answer and explanation

Correct option: B

\frac{3}{(1-x)^2}

Option B (

\frac{3}{(1-x)^2}

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

Revision notes and key takeaways

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