If ${y}={A} \sin \omega {t}$ then $\frac{d^5 y}{d t^5}=$

$A \omega^5 \cos \left(\omega t-\frac{\pi}{2}\right)$

$A \omega^5 \cos \left(\omega t+\frac{\pi}{2}\right)$

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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Mixed difficulty

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781 Easy

781 of 932

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

782 Easy

782 of 932

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.

783 Easy

783 of 932

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).

784 Medium

784 of 932

Application +4 | -1

{y}={A} \sin \omega {t}

then

\frac{d^5 y}{d t^5}=

View answer and explanation

Correct option: D

A \omega^5 \sin \left(\omega t+\frac{\pi}{2}\right)

Option D is correct because the fifth derivative of y = A sin(ωt) is Aω^5 sin(ωt + π/2), which can be found by repeatedly differentiating the function.

785 Easy

785 of 932

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}}}

786 Medium

786 of 932

Application +4 | -1

y=\sin (\sin x)

, and

\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x} \tan x+f(x)=0

, then

f(x)

View answer and explanation

Correct option: D

\cos ^{2} x \sin (\sin x)

The correct answer is D (

\cos ^{2} x \sin (\sin x)

) because it is the function f(x) that satisfies the given differential equation after finding the first and second derivatives of y.

787 Easy

787 of 932

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

788 Easy

788 of 932

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.

789 Easy

789 of 932

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.

790 Medium

790 of 932

Application +4 | -1

x = a{\cos ^4}\mathrm{\theta },y = a{\sin ^4}\mathrm{\theta },

then

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

\mathrm{\theta } = \frac{{3\pi }}{4}

View answer and explanation

Correct option: A

- 1

Option A (-1) is correct because after finding dy/dx in terms of theta, substituting theta = 3pi/4 yields -1.

791 Medium

791 of 932

Application +4 | -1

f(x)

is given by,

f(x)=(\cos x+i \sin x)

(\cos 3 x+i \sin 3 x) \ldots(\cos (2 n-1) x+i \sin (2 n-1) x)

, then

f^{\prime \prime}(x)

is equal to

View answer and explanation

Correct option: B

-n^4 f(x)

Option B (-n⁴ f(x)) is correct because it is the second derivative of the given function, calculated using Euler's formula and differentiation.

792 Easy

792 of 932

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.

793 Easy

793 of 932

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.

794 Easy

794 of 932

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.

795 Medium

795 of 932

Analyzing +4 | -1

Let

f

be a differentiable function from

R

R

such that

|f(x) - f(y)| \le 2|x - y{|^{3/2}}\mathrm{ }\,\mathrm{for}\,\,\mathrm{ all}\,\mathrm{ }x,y \in R\mathrm{. }

f(0)=1

, then

\int_0^1 f^2(x) d x

is equal to

View answer and explanation

Correct option: D

Option D (1) is correct because the given condition implies that the derivative of

f(x)

is zero everywhere, making

f(x)

a constant function. Since

f(0)=1

f(x)=1

for all

x

, and the integral of

f^2(x)

from 0 to 1 is 1.

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796 Medium

796 of 932

Application +4 | -1

f(x)

is differentiable at

x=a

, then

\lim\limits _{x \rightarrow a} \frac{x^2 f(a)-a^2 f(x)}{x-a}

is equal to

View answer and explanation

Correct option: C

2 a f(a)-a^2 f^{\prime}(a)

Option C (

2 a f(a)-a^2 f^{\prime}(a)

) is correct because it is the result of applying the definition of the derivative and algebraic manipulation to evaluate the limit.

797 Easy

797 of 932

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.

798 Medium

798 of 932

Application +4 | -1

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

then

\frac{d y}{d x}=

View answer and explanation

Correct option: B

\frac{y}{x}

Option B (

\frac{y}{x}

) is correct because implicit differentiation and simplification lead to this result.

799 Easy

799 of 932

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.

800 Medium

800 of 932

Analyzing +4 | -1

Let

f(x)=15-|x-10| ; x \in R

. Then the set of all values of

x

, at which the function,

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

is not differentiable, is:

View answer and explanation

Correct option: A

\{5,10,15\}

Option A is correct because the points where g(x) is non-differentiable are the points where the absolute value functions inside g(x) change sign, which are 5, 10, and 15.

801 Easy

801 of 932

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

802 Medium

802 of 932

Application +4 | -1

The rate of change of

\sqrt {\left( {{x^2} + 16} \right)}

with respect to

\frac{x}{{x - 1}}

x = 3

View answer and explanation

Correct option: C

- \frac{{12}}{5}

Option C (

- \frac{{12}}{5}

) is correct because the rate of change is calculated by finding the derivatives of both functions and then dividing them, evaluated at x=3.

803 Medium

803 of 932

Application +4 | -1

\frac{d}{d x}\left(\log \left|\tan \frac{x}{2}\right|\right)

is equal to

View answer and explanation

Correct option: C

\operatorname{cosec} x

Option C (

\operatorname{cosec} x

) is correct because by applying the chain rule and simplifying the derivative of

\log |\tan \frac{x}{2}|

, we obtain

\operatorname{cosec} x

804 Medium

804 of 932

Application +4 | -1

f(x)=|\sin x-| \cos x||

, then

f^{\prime}\left(\frac{7 \pi}{6}\right)=

View answer and explanation

Correct option: C

\frac{\sqrt{3}-1}{2}

Option C (

\frac{\sqrt{3}-1}{2}

) is correct because it is the value of the derivative of the function at the given point after considering the absolute values.

805 Easy

805 of 932

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.

806 Medium

806 of 932

Application +4 | -1

y=2^{\log x}

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: D

\frac{2^{\log x} \cdot \log 2}{x}

Option D (

\frac{2^{\log x} \cdot \log 2}{x}

) is correct because it is the result of applying the chain rule and derivative of exponential functions.

807 Medium

807 of 932

Application +4 | -1

Let

y

be an implicit function of

x

defined by

{x^{2x}} - 2{x^x}\cot y - 1 = 0.

Then

y'\left( 1 \right)

equals

View answer and explanation

Correct option: A

- 1

Option A (-1) is correct because it is the value of y'(1) obtained by implicit differentiation and substituting x=1 and y=pi/2.

808 Medium

808 of 932

Analyzing +4 | -1

Let

f\left( x \right) = {x^3} - 6{x^2} + 15x + 3.

Then.

View answer and explanation

Correct option: C

f\left( x \right)

is invertible

Option C (

f(x)

is invertible) is correct because the derivative is always positive, indicating that the function is strictly increasing and therefore invertible.

809 Easy

809 of 932

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.

810 Easy

810 of 932

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.

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