If $y=\sqrt{x+\sqrt{x+\sqrt{x}}}, x > 0$, then $d y / d x$ at $x=1$ is

$\frac{1}{\sqrt{1+\sqrt{2}}}\left(\frac{3+4 \sqrt{2}}{4 \sqrt{2}}\right)$

$\left(\frac{3+4 \sqrt{2}}{8 \sqrt{2}}\right)$

$\frac{3+4 \sqrt{2}}{\sqrt{1+\sqrt{2}}}$

$y y^{\prime \prime}-2\left(y^{\prime}\right)^2+1=0$

$y y^{\prime \prime}+2\left(y^{\prime}\right)^2+1=0$

Differential coefficient of $\sqrt {\sec \sqrt x } $ is

$\frac{1}{{4\sqrt x }}\sec \sqrt x \sin \sqrt x $

$\frac{1}{2}\sqrt x \sec \sqrt x \sin \sqrt x $

$\frac{1}{2}\sqrt x {\left( {\sec \sqrt x } \right)^{\frac{3}{2}}}.\sin \sqrt x $

Back to DIFFERENTIATION

Your Progress

0 / 500 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • DIFFERENTIATION

Differentiation practice · Easy

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

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.

Take a chapter quiz →

Showing 91-120 of 500 questions

Preparing practice and saved progress…

91 Easy

91 of 500

Application +4 | -1

2f\left( x \right) = f'\left( x \right)

and

f\left( 0 \right) = 3,

then

f\left( 2 \right)

is equal to

View answer and explanation

Correct option: A

3{e^4}

Option A (

3{e^4}

) is correct because it satisfies the given differential equation and initial condition.

92 Easy

92 of 500

Application +4 | -1

y=x^{x^{x^{x^{x^{...}}}}}

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: C

\frac{y}{x(1+y \log x)}

Option B is correct because it is derived by taking the logarithm of both sides and then differentiating implicitly.

93 Easy

93 of 500

Application +4 | -1

y=\sqrt{x+\sqrt{x+\sqrt{x}}}, x > 0

, then

d y / d x

x=1

View answer and explanation

Correct option: A

\frac{1}{2 \sqrt{1+\sqrt{2}}}\left(\frac{3+4 \sqrt{2}}{4 \sqrt{2}}\right)

Option A is correct because it is the result of applying the chain rule multiple times and evaluating the derivative at x=1.

94 Easy

94 of 500

Application +4 | -1

y=\cot ^{-1}\left(x^2\right)

, then

\frac{d y}{d x}

is equal to :

View answer and explanation

Correct option: C

\frac{-2 x}{1+x^4}

Option C (

\frac{{-2 x}}{{1+x^4}}

) is correct because it is the derivative of arccot(x²) using the chain rule.

95 Easy

95 of 500

Application +4 | -1

\frac{d}{d x} \log (\log x)

is equal to

View answer and explanation

Correct option: C

(x \log x)^{-1}

Option C (

(x \log x)^{-1}

) is correct because the derivative of

\log(\log x)

\frac{1}{\log x} \cdot \frac{1}{x}

, which simplifies to

(x \log x)^{-1}

96 Easy

96 of 500

Application +4 | -1

y=\sin ^{-1}\left(\frac{\sin \alpha \sin x}{1-\cos \alpha \sin x}\right)

, then

y^{\prime}(0)

View answer and explanation

Correct option: D

\sin \alpha

Option D (

\sin \alpha

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

y

with respect to

x

evaluated at

x=0

97 Easy

97 of 500

Application +4 | -1

Let

f\left( x \right) = \frac{1}{{\sqrt {18 - {x^2}} }}

. The value of

\lim\limits_{x \to 3} \frac{{f\left( x \right) - f\left( 3 \right)}}{{x - 3}}

, is

View answer and explanation

Correct option: D

\frac{1}{9}

Option D (

\frac{1}{9}

) is correct because the limit is the derivative of

f(x)

evaluated at

x=3

, which equals

\frac{1}{9}

98 Easy

98 of 500

Application +4 | -1

f\left( x \right) = \sqrt {1 + {{\cos }^2}\left( {{x^2}} \right)} ,

then

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

View answer and explanation

Correct option: B

- \sqrt {\frac{\pi }{6}}

Option B (

- \sqrt {\frac{\pi }{6}}

) is correct because it's the result of applying the chain rule and evaluating the derivative at the given point.

99 Easy

99 of 500

Application +4 | -1

If a function is represented parametrically by the equations

x=\frac{1+t}{t^3}, \quad y=\frac{3}{2 t^2}+\frac{2}{t}

then

View answer and explanation

Correct option: B

x\left(\frac{d y}{d x}\right)^3=1+\frac{d y}{d x}

Option B (

x(\frac{{d y}}{{d x}})^3=1+\frac{{d y}}{{d x}}

) is correct because after finding

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

and substituting in the given equation, it satisfies the equation.

100 Easy

100 of 500

Application +4 | -1

2^x+2^y=2^{x+y}

, then the value of

\frac{d y}{d x}

x=y

=1

View answer and explanation

Correct option: B

-1

Option B (-1) is correct because after differentiating the given equation and substituting x=y=1, the value of dy/dx is -1.

101 Easy

101 of 500

Analyzing +4 | -1

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

, then

\frac{d y}{d x}

is equal to

View answer and explanation

Correct option: D

None of these

The correct answer is D (None of these) because the derivative calculated does not match any of the given options.

102 Easy

102 of 500

Analyzing +4 | -1

Let

f(x) = {\log _e}(\sin x),(0 < x < \pi )

and

g(x) = {\sin ^{ - 1}}({e^{ - x}}),(x \ge 0)

. If

\alpha

is positive real number
such that

a = (fog)'(\alpha )

and

b = (fog)(\alpha )

, then

View answer and explanation

Correct option: B

a{\alpha ^2} - b\alpha - a = 1

Option B (

a{\alpha ^2} - b\alpha - a = 1

) is correct because the given functions are composed and simplified, then differentiated and evaluated at

\alpha

to satisfy the equation.

103 Easy

103 of 500

Application +4 | -1

\frac{d}{d x}\left(\frac{\sin x}{1+\cos x}\right)=

View answer and explanation

Correct option: A

\frac{1}{2} \sec ^{2} \frac{x}{2}

Option A (1/2 sec²(x/2)) is correct because the derivative of sin(x)/(1+cos(x)) simplifies to tan(x/2), and the derivative of tan(x/2) is 1/2 sec²(x/2).

104 Easy

104 of 500

Analyzing +4 | -1

The derivative of

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

with respect to

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

x = \frac{1}{2}

View answer and explanation

Correct option: D

\frac{{\sqrt 3 }}{{10}}\,

Option D is correct because after simplification using trigonometric substitutions and differentiation, the derivative evaluates to

\frac{{\sqrt 3 }}{{{{10}}}}

x = \frac{1}{2}

105 Easy

105 of 500

Application +4 | -1

f(x)=\log _x\{\ln (x)\}

, then

f^{\prime}(x)

x=e

, is

View answer and explanation

Correct option: D

e^{-1}

Option D (

e^{-1}

) is correct because after applying the change of base formula and differentiating, the value of the derivative at x=e is 1/e.

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

106 Easy

106 of 500

Application +4 | -1

f(x)=\cot ^{-1}\left(\frac{x^x-x^{-x}}{2}\right)

, then

f^{\prime}(1)

equals

View answer and explanation

Correct option: A

-1

Option A (-1) is correct because it is the value of the derivative of the given function, evaluated at x=1.

107 Easy

107 of 500

Application +4 | -1

x=a\left[\cos \theta+\log \tan \frac{\theta}{2}\right],

y=a \sin \theta

then

\frac{d y}{d x}=

View answer and explanation

Correct option: C

\tan \theta

Option C (

\tan \theta

) is correct because it is the result of calculating

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

using parametric differentiation.

108 Easy

108 of 500

Application +4 | -1

f\left( x \right) = {\log _a}\left( {{{\log }_a}x} \right)

, then

f'\left( x \right)

View answer and explanation

Correct option: A

\frac{{{{\log }_a}e}}{{x{{\log }_e}x}}

Option A (

\frac{{{{\log }_a}e}}{{x{{\log }_e}x}}

) is correct because it is the correct derivative of the given logarithmic function using the chain rule and change of base formula.

109 Easy

109 of 500

Application +4 | -1

x^2+y^2=1

, then

View answer and explanation

Correct option: B

y y^{\prime \prime}+\left(y^{\prime}\right)^2+1=0

Option B (y y'' + (y')² + 1 = 0) is correct because it is the result of differentiating the given equation twice and simplifying.

110 Easy

110 of 500

Application +4 | -1

y=2^{1 / \log _{x} 4}

, then

x

is equal to

View answer and explanation

Correct option: C

y^{2}

Option C is correct because it is the result of simplifying the expression using properties of logarithms.

111 Easy

111 of 500

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

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

The correct answer is A (

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

) because it is the result of differentiating the given expression after simplifying it.

112 Easy

112 of 500

Application +4 | -1

Let

f(x)

be a differentiable function such that

f^{\prime}(x)=\sin x+\sin 4 x \cos x

. Then,

f^{\prime}\left(2 x^2+\frac{\pi}{2}\right)

x=\sqrt{\frac{\pi}{2}}

is equal to

View answer and explanation

Correct option: C

-2 \sqrt{2 \pi}

Option C is correct because it correctly applies the chain rule and trigonometric identities to find the derivative and evaluate it at the given point.

113 Easy

113 of 500

Analyzing +4 | -1

y^{1 / m}=\left[x+\sqrt{1+x^2}\right]

, then

\left(1+x^2\right) y_2+x y_1

is equal to

View answer and explanation

Correct option: A

m^2 y

Option A (

m^2 y

) is correct because it is the result of differentiating the given equation twice and simplifying using the given relation.

114 Easy

114 of 500

Analyzing +4 | -1

y = {\tan ^{ - 1}}\left( {\frac{{\log \left( {\frac{e}{{{x^2}}}} \right)}}{{\log \left( {e{x^2}} \right)}}} \right) + {\tan ^{ - 1}}\left( {\frac{{3 + 2\log x}}{{1 - 6\log x}}} \right)

, then

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

View answer and explanation

Correct option: C

Option C (0) is correct because after simplifying the expression for y, we find that y is a constant, and therefore its second derivative is 0.

115 Easy

115 of 500

Application +4 | -1

Differential coefficient of

\sqrt {\sec \sqrt x }

View answer and explanation

Correct option: B

\frac{1}{{4\sqrt x }}{\left( {\sec \sqrt x } \right)^{\frac{3}{2}}}.\sin \sqrt x

Option B (

\frac{1}{{4\sqrt x }}{\left( {\sec \sqrt x } \right)^{\frac{3}{2}}}.\sin \sqrt x

) is correct because it correctly applies the chain rule to find the derivative of the given function.

116 Easy

116 of 500

Application +4 | -1

f

and

g

are differentiable functions such that

g^{\prime}(a)=2

and

g(a)=b

and if

f \circ g

is an identity function, then

f^{\prime}(b)

has the value equal to

View answer and explanation

Correct option: D

1 / 2

Option D (

\frac{1}{2}

) is correct because it is the value of f'(b) calculated using the properties of inverse functions and the chain rule.

117 Easy

117 of 500

Application +4 | -1

If the parametric equation of a curve given by

x = {e^t}\;\mathrm{cost},\;y = {e^t}\sin t

then the tangent to the curve at the point

t = \pi /4

makes with axis of

x

the angle

View answer and explanation

Correct option: D

\frac{\pi }{2}

Option D (

\frac{\pi}{2}

) is correct because the tangent at t =

\frac{\pi}{4}

has an infinite slope, indicating a vertical line.

118 Easy

118 of 500

Application +4 | -1

x=\cos \theta, y=\sin ^3 \theta

, then

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

\theta=\pi / 2

View answer and explanation

Correct option: D

-3

Option D (-3) is correct because after finding the first and second derivatives and substituting

\theta = \frac{{\pi}}{{2}}

, the expression evaluates to -3.

119 Easy

119 of 500

Application +4 | -1

Let

f

and

g

be differentiable functions satisfying

g'\left( a \right) = 2,g\left( a \right) = b

and

fog = I

(identify function). Then,

f'\left( b \right)

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}

Option A (1/2) is correct because, given fog = I, the derivative f'(b) can be found using the chain rule and the fact that g'(a) = 2 and g(a) = b.

120 Easy

120 of 500

Application +4 | -1

y^x-x^y=1

, then the value of

\frac{d y}{d x}

x=1

View answer and explanation

Correct option: A

2(1-\log 2)

Option A (

2(1-\log 2)

) is correct because it is the result of differentiating the given equation and substituting the given values.

Revision notes and key takeaways

Review DIFFERENTIATION in your textbook or class notes, then use this set to check your understanding.

Showing 91-120 of 500 questions

Your Progress

Differentiation practice · Easy

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice