If $ y=\sqrt{\frac{1+e^{x}}{1-e^{x}}} $, then $ \frac{d y}{d x} $ is equal to

$ \frac{e^{x}}{\left(1-e^{x}\right) \sqrt{1-e^{x}}} $

$ \frac{e^{x}}{\left(1-e^{x}\right) \sqrt{1+e^{x}}} $

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JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Mixed difficulty

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871 Medium

871 of 932

Understanding +4 | -1

Suppose

f(x)

is differentiable for all

x

and

\lim\limits _{h \rightarrow 0} \frac{1}{h} f(1+h)=5

, then

f^{\prime}

(1) equals

View answer and explanation

Correct option: B

Option B (5) is correct because

f'(1) = \lim\limits_{h \to 0} \frac{{f(1 + h) - f(1)}}{h}

and given

\lim\limits _{h \rightarrow 0} \frac{1}{h} f(1+h)=5

, it implies

f(1)=0

and

f'(1)=5

872 Medium

872 of 932

Application +4 | -1

8f\left( x \right) + 6f\left( {\frac{1}{x}} \right) = x + 5

and

y = {x^2}f\left( x \right),

then

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

x = - 1

View answer and explanation

Correct option: C

- \frac{1}{{14}}

Option C (

\frac{{ - 1}}{{14}}

) is correct because it is the value of

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

x = -1

after solving for

f(x)

and differentiating

y = x^2 f(x)

873 Hard

873 of 932

Analyzing +4 | -1

Let

f\left( x \right)

be defined on

R

such that

f\left( 1 \right) = 2,f\left( 2 \right) = 8

and

f\left( {u + \upsilon } \right) = f\left( u \right) + ku\upsilon - 2{\upsilon ^2}

for all

u,\upsilon \in R

(

k

is a fixed constant). Then,

View answer and explanation

Correct option: A

f'\left( x \right) = 8x

Option A (

f'(x) = 8x

) is correct because by substituting

u = x

and

\upsilon = h

into the given functional equation and taking the limit as

h

approaches 0, we find the derivative.

874 Easy

874 of 932

Analyzing +4 | -1

{x^y} = {y^x},

then

x(x - y\log x)\frac{{dy}}{{dx}}

is equal to

View answer and explanation

Correct option: A

y(y - x\log y)

Option A (

y(y - x\log y)

) is correct because after differentiating the given equation and rearranging, the expression simplifies to the stated result.

875 Easy

875 of 932

Application +4 | -1

A man who is

1.5 \mathrm{~m}

tall walks away from a lamp which is

3 \mathrm{~m}

above the ground at the rate of

30 \mathrm{~m} / \mathrm{min}

the shadow lengthening at

\mathrm{K} \mathrm{m} / \mathrm{min}

. Then

\mathrm{K}=

View answer and explanation

Correct option: D

Option D (30) is correct because the rate at which the shadow lengthens is equal to the rate at which the man walks away from the lamp.

876 Medium

876 of 932

Application +4 | -1

f\left(\frac{3 x-4}{3 x+4}\right)=x+2

then

\int f(x) d x

is equal to

View answer and explanation

Correct option: B

-\frac{8}{3} \ln |(1-x)|+\frac{2}{3} x+c

Option B (

-\frac{8}{3} \ln |(1-x)|+\frac{2}{3} x+c

) is correct because after substituting and integrating, the result matches this option.

877 Medium

877 of 932

Application +4 | -1

f:\left( { - 1,1} \right) \to R

be a differentiable function with

f\left( 0 \right) = - 1

and

f'\left( 0 \right) = 1.

Let

g(x) = {\left[ {f\left( {2f\left( x \right) + 2} \right)} \right]^2}

. Then,

\mathrm{g'}\left( 0 \right)

is equal to

View answer and explanation

Correct option: A

-4

Option A (-4) is correct because after applying the chain rule and substituting

x=0

, we find that

g'(0) = -4

878 Easy

878 of 932

Analyzing +4 | -1

Observe the following statements:

I. If

f\left( x \right) = a{x^{41}} + b{x^{ - 40}}

, then

\frac{{f''\left( x \right)}}{{f\left( x \right)}} = 1640{x^{ - 2}}

II.

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

Which of the following is correct?

View answer and explanation

Correct option: A

I is true, but II is false

Option A (I is true, but II is false) is correct because statement I is verified to be true through differentiation, while statement II is false due to the range of the inverse tangent function.

879 Medium

879 of 932

Analyzing +4 | -1

Let

f\left( {\frac{{x + y}}{2}} \right) = \frac{{f(x) + f(y)}}{2}

and

f'(0) = a\mathrm{ \,\,and}\,\,\,\mathrm{ }f(0) = b.

Find

f''(x)

(where

y

is independent of

x

), when

f(x)

is differentiable.

View answer and explanation

Correct option: A

Option A (0) is correct because the second derivative of f(x) is 0, indicating that f(x) is a linear function.

880 Medium

880 of 932

Application +4 | -1

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

, then

\frac{d y}{d x}=

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because after substituting

x = \sin \theta

and simplifying the expression, the derivative is found to be

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

881 Easy

881 of 932

Analyzing +4 | -1

f(x)=\frac{a+\sqrt{a^2-x^2}+x}{\sqrt{a^2-x^2}+a-x}

, where

a>0

and

x < a

, then

f'(0)

has the value equal to

View answer and explanation

Correct option: D

\frac{1}{a}

Option D (

\frac{1}{a}

) is correct because after simplifying the function and finding its derivative, the value of

f'(0)

\frac{1}{a}

882 Medium

882 of 932

Application +4 | -1

Let

f(x)=2^{x(x-1)}

for all

x \geq 1

. Then

\left(f^{-1}\right)^{\prime}(4)

(1 / k) \log _2 e

where the value of

k

View answer and explanation

Correct option: D

Option D (12) is correct. The derivative of the inverse function is calculated and evaluated at x=4, leading to k=12.

883 Medium

883 of 932

Application +4 | -1

y=\sin ^{-1} \frac{\sqrt{(1+x)}+\sqrt{(1-x)}}{2}

, then

\frac{d y}{d x}=

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it is the result of simplifying the expression using trigonometric substitution and then differentiating.

884 Easy

884 of 932

Application +4 | -1

The derivative of

{\cos ^3}x

w.r.t.

{\sin ^3}x

View answer and explanation

Correct option: A

- \cot x

Option A (

- \cot x

) is correct because it is the result of dividing the derivative of

{\cos ^3}x

by the derivative of

{\sin ^3}x

885 Easy

885 of 932

Application +4 | -1

y=\left(x^2+1\right)^{\sin x}

, then

y^{\prime}(0)

is equal to

View answer and explanation

Correct option: C

Option C (0) is correct because differentiating

y

and substituting

x=0

results in

y'(0) = 0

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886 Medium

886 of 932

Application +4 | -1

Let

f:(-1,1) \rightarrow R

be a differentiable function with

f(0)=1

and

f^{\prime}(0)=-1.

g(x)=(f(n f(x)-n))^n

, where

n

is a natural number and

g^{\prime}(0)= 4,

then

n

equals

View answer and explanation

Correct option: A

Option A is correct because solving for n in the given conditions leads to n = 2 after applying the chain rule and substituting the given values.

887 Medium

887 of 932

Analyzing +4 | -1

Let for

\mathrm{i}=1,2,3, \mathrm{p}_{\mathrm{i}}(\mathrm{x})

be a polynomial of degree 2 in

\mathrm{x}, \mathrm{p}_{\mathrm{i}}^{\prime}(\mathrm{x})

and

\mathrm{p}_{\mathrm{\prime \prime}}{ }_{\mathrm{\prime}}(\mathrm{x})

be the first and second order derivatives of

\mathrm{p}_{\mathrm{i}}(\mathrm{x})

respectively. Let,

A(x)=\left[\begin{array}{lll} \mathrm{p}_{1}(\mathrm{x}) & \mathrm{p}_{1}^{\prime}(\mathrm{x}) & \mathrm{p}_{1}^{\prime \prime}(\mathrm{x}) \\ \mathrm{p}_{2}(\mathrm{x}) & \mathrm{p}_{2}{ }^{\prime}(\mathrm{x}) & \mathrm{p}_{2}^{\prime \prime}(\mathrm{x}) \\ \mathrm{p}_{3}(\mathrm{x}) & \mathrm{p}_{3}{ }^{\prime}(\mathrm{x}) & \mathrm{p}_{3}{ }^{\prime \prime}(\mathrm{x}) \end{array}\right]

and

\mathrm{B}(\mathrm{x})=[\mathrm{A}(\mathrm{x})]^{T} \mathrm{A}(\mathrm{x})

. Then determinant of

\mathrm{B}(\mathrm{x})

View answer and explanation

Correct option: A

is a polynomial of degree 6 in

x

Option A is correct. The determinant of B(x) is a polynomial of degree 6 because the elements of B(x) are at most degree 4, and the determinant involves products of these elements.

888 Easy

888 of 932

Application +4 | -1

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

, then

f^{\prime}\left(\frac{\pi}{4}\right)

is equal to

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the left-hand derivative and right-hand derivative at x = pi/4 are not equal, so the derivative does not exist.

889 Easy

889 of 932

Analyzing +4 | -1

The derivative of the function,

f(x) = {\cos ^{ - 1}}\left\{ {\frac{1}{{\sqrt {13} }}} \right.(2\cos x - 3\sin x)\} \mathrm{ } + {\sin ^{ - 1}}\left\{ {\frac{1}{{\sqrt {13} }}(2\cos x + 3\sin x)} \right\}

\mathrm{ w}\mathrm{.r}\mathrm{.t}\mathrm{.}\,\,\,\sqrt {1 + {x^2}} \,\,\,\mathrm{at}\,\,\,x = \frac{3}{4}\,\,\mathrm{is}

View answer and explanation

Correct option: C

\frac{10}{3}

Option C (

\frac{10}{3}

) is correct because after simplification and differentiation, the value of the derivative at

x = \frac{3}{4}

\frac{10}{3}

890 Medium

890 of 932

Application +4 | -1

A function

y=f(x)

satisfies

x \frac{d y}{d x}=\sqrt{y^2+1}, x>0 \& f(2)=0

, then

f(4)=

View answer and explanation

Correct option: B

0.75

Option B (0.75) is correct because solving the differential equation and applying the initial condition f(2)=0 yields f(4)=0.75.

891 Easy

891 of 932

Application +4 | -1

y=\sqrt{\frac{1+e^{x}}{1-e^{x}}}

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: A

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

Option A is correct because the derivative is found using the chain rule and quotient rule, simplifying to the given expression.

892 Medium

892 of 932

Analyzing +4 | -1

z = \tan \left( {y + ax} \right) + \sqrt {y - ax} \Rightarrow {z_{xx}} - {a^2}{z_{yy}}

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because after finding the second-order partial derivatives and substituting them into the expression, the result is zero.

893 Medium

893 of 932

Application +4 | -1

z = {\sec ^{ - 1}}\left( {\frac{{{x^4} + {y^4} - 8{x^2}{y^2}}}{{{x^2} + {y^2}}}} \right),

then

x\frac{{\partial z}}{{\partial x}} + y\frac{{\partial z}}{{\partial y}}

is equal to

View answer and explanation

Correct option: B

2\cot z

Option B (

2\cot z

) is correct because the given function is homogeneous and Euler's theorem can be applied to find the value of the expression.

894 Easy

894 of 932

Application +4 | -1

Let

f^{\prime}(x)=\sin \left(x^2\right)

and

y=f\left(x^2+1\right)

then

\frac{d y}{d x}

x=1

is :

View answer and explanation

Correct option: C

2 \sin 4

Option C (

2 \sin 4

) is correct because it is the result of applying the chain rule and evaluating the derivative at x=1.

895 Medium

895 of 932

Application +4 | -1

g

is inverse of

f

and

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

then

g^{\prime}(x)

equals

View answer and explanation

Correct option: C

1+[g(x)]^n

Option C (

1+[g(x)]^n

) is correct because it is derived from the relationship between the derivatives of a function and its inverse.

896 Easy

896 of 932

Analyzing +4 | -1

Let

f(x)=\left\{\begin{array}{cc}(x-1) \sin \frac{1}{x-1} & \text { if } x \neq 1 \\ 0 & \text { if } x=1\end{array}\right.

Then which one of the following is true?

View answer and explanation

Correct option: B

f

is differentiable at

x=0

but not at

x=1

Option B is correct. The function is differentiable at x=0 because the derivative exists at that point, but it is not differentiable at x=1 because the limit does not exist.

897 Easy

897 of 932

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.

898 Easy

898 of 932

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

899 Medium

899 of 932

Application +4 | -1

f(x)=\sin ^{-1}\left(\frac{2 \times 3^{x}}{1+9^{x}}\right)

, then

f^{\prime}\left(-\frac{1}{2}\right)

equals.

View answer and explanation

Correct option: A

\sqrt{3} \log _{e} \sqrt{3}

Option A (

\sqrt{3} \log _{e} \sqrt{3}

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

900 Easy

900 of 932

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.

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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