Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 4 - Methods of Differentiation>DPP>Q 8

EASY

JEE Main

IMPORTANT

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Let $g (x) = f (x) \sin x$ , where $f (x)$ is a twice differentiable function on $(- \infty, \infty)$ such that $f^{'} (- π) = 1$ . Then the value of $g^{"} (- π)$ equals

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Important Questions on Methods of Differentiation

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 4 - Methods of Differentiation>DPP>Q 9

If

f (x) = \log_{x} (\log_{e} x)

, then

f^{'} (x)

at

x = e

is:

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 4 - Methods of Differentiation>DPP>Q 10

Let

g (x) = e^{f (x)}

and

f (x + 1) = x + f (x) \forall x \in R

. If

n \in I^{+},

then find the value of

\frac{g^{'} (n + \frac{1}{2})}{g (n + \frac{1}{2})} - \frac{g^{'} (\frac{1}{2})}{g (\frac{1}{2})}

.

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 4 - Methods of Differentiation>DPP>Q 11

Find the value of

\frac{d}{d x} [\cos^{- 1} (x \sqrt{x} - \sqrt{(1 - x) (1 - x^{2})})]

.

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 4 - Methods of Differentiation>DPP>Q 12

If

y = \cos^{- 1} \sqrt{\frac{\sqrt{1 + x^{2}} + 1}{2 \sqrt{1 + x^{2}}}}

, then

\frac{d y}{d x}

is equal to:

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 4 - Methods of Differentiation>DPP>Q 13

Suppose that

f (x)

is a differentiable invertible function such that

f^{'} (x) \neq 0

and

h^{'} (x) = f (x) .

Given that

f (1) = f^{'} (1) = 1, h (1) = 0

and

g (x)

is the inverse of

f (x)

. Let

G (x) = x^{2} g (x) - x h (g (x)), \forall x \in R

. Which of the following is/are correct?

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 4 - Methods of Differentiation>DPP>Q 1

If

t (1 + x^{2}) = x

and

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

, then at

x = 2,

the value of

\frac{d y}{d x}

is:

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 4 - Methods of Differentiation>DPP>Q 2

If

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

satisfy

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

, then

f (x)

equals:

MEDIUM

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 4 - Methods of Differentiation>DPP>Q 3

Let

y = x^{3} - 8 x + 7

and

x = f (t)

. If

\frac{d y}{d t} = 2

and

x = 3

at

t = 0,

then

\frac{d x}{d t}

at

t = 0

is