Method of Differentiation — JEE Maths Practice Questions & Solutions

hard

If

y=\dfrac{x}{\sqrt{a^{2}-1}}-\dfrac{2}{\sqrt{a^{2}-1}}\tan^{-1}\!\left(\dfrac{\sin x}{a+\sqrt{a^{2}-1}+\cos x}\right)

, where

a\in(-\infty,-1)\cup(1,\infty)

, then

y'\!\left(\dfrac{\pi}{2}\right)

equals

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hard

If

f(x)=(2015+x)^{n}

where

x

is a real variable and

n

is a positive integer, then the value of

f(0)+\dfrac{f'(0)}{1!}+\dfrac{f''(0)}{2!}+\cdots+\dfrac{f^{(n-1)}(0)}{(n-1)!}

is

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hard

If

f''(x)=e^{x}\cos x

,

\displaystyle\lim_{x\to-\infty}f'(x)=0

and

y=f\!\left(\dfrac{x-1}{x+1}\right)

, then

\dfrac{dy}{dx}

at

x=1

is

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hard

If

y=\dfrac{\alpha x+\beta}{\gamma x+\delta}

, then

2\dfrac{dy}{dx}\cdot\dfrac{d^{3}y}{dx^{3}}

is

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medium

If

y = x\sin(\log x) + x\log x

, then

x^2\dfrac{d^2 y}{dx^2} - x\dfrac{dy}{dx} + 2y

is:

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medium

If

y=\dfrac{1}{1+x^{n-m}+x^{p-m}}+\dfrac{1}{1+x^{m-n}+x^{p-n}}+\dfrac{1}{1+x^{m-p}+x^{n-p}}

, then

\dfrac{dy}{dx}

at

x=e^{mnp}

is equal to

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medium

If

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

, then

f'(x)

at

x=e

is

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medium

Let

f(x)

be a polynomial function of second degree. If

f(1)=f(-1)

and

a,b,c

are in A.P., then

f'(a),f'(b),f'(c)

are in

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medium

If

u=f(\tan x)

,

v=g(\sec x)

and

f'(x)=\tan^{-1}x

,

g'(x)=\csc^{-1}x

, then

\dfrac{du}{dv}

at

x=\dfrac{\pi}{4}

is

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medium

If

f(x)=(1+x)(1+x^{2})(1+x^{4})\cdots(1+x^{2^{n}})

, then at

x=0

,

f'(x)

is

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medium

\dfrac{d}{dx}\left[\cos^{-1}\!\left(x\sqrt{x}-\sqrt{(1-x)(1-x^{2})}\right)\right]

is

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medium

If

y=\dfrac{1}{1+x+x^{2}+x^{3}}

, then

\left(\dfrac{dy}{dx}\right)_{x=0}

and

\left(\dfrac{d^{2}y}{dx^{2}}\right)_{x=0}

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medium

If

5f(x)+3f\!\left(\dfrac{1}{x}\right)=x+2

and

y=xf(x)

, then

\dfrac{dy}{dx}

at

x=1

is

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medium

If

y=f\!\left(\dfrac{2x-1}{x^{2}+1}\right)

and

f'(x)=\sin x^{2}

, then

y'(0)

is

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medium

Let

f_{n+1}(x) = e^{f_n(x)}

for all

1 \le n \le 9

and

f_1(x) = e^x

. Then the first derivative of

f_n(x)

for

n = 10

is:

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medium

If

F(x) = f(x)\phi(x)

and

f'(x)\phi'(x) = c

, then

\dfrac{f''(x)}{f(x)} + \dfrac{\phi''(x)}{\phi(x)} + \dfrac{2c}{f(x)\phi(x)}

is:

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medium

Let

y = f(x)

be a real valued twice differentiable function defined on

\mathbb{R}

. Then

\dfrac{d^2 y}{dx^2}\left(\dfrac{dx}{dy}\right)^3 + \dfrac{d^2 x}{dy^2}

is

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medium

If

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

, then

(1 - x^2)\dfrac{d^2 y}{dx^2} - x\dfrac{dy}{dx}

is:

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medium

Let

f(x)

and

g(x)

be two functions having finite non-zero third order derivatives

f'''(x)

and

g'''(x)

for all

x \in \mathbb{R}

. If

f(x)\, g(x) = 1

for all

x \in \mathbb{R}

, then

\dfrac{f'''}{f'} - \dfrac{g'''}{g'}

is equal to:

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medium

If

y^{1/m} = x + \sqrt{1 + x^2}

, then

(1 + x^2)y_2 + x y_1

is equal to:

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medium

If

f(x) = 4x^5 + 3x^3 + 2x^2 + e^{x/7}

and

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

, then

g'(1)

is.

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medium

If

x + \cos\theta = \sec\theta

and

y + \cos^8\theta = \sec^8\theta

, then

\left(\dfrac{x^2 + 4}{y^2 + 4}\right)\left(\dfrac{dy}{dx}\right)^2

is:

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medium

Let

f

and

g

be two real-valued differentiable functions on

\mathbb{R}

. If

f'(x) = g(x)

and

g'(x) = f(x)

for all

x \in \mathbb{R}

, and

f(3) = 5

,

g(3) = 4

, then the value of

f^2(\pi) - g^2(\pi)

.

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medium

If

xy = (x+y)^n

and

\dfrac{dy}{dx} = \dfrac{y}{x}

, then

n

is.

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medium

Suppose the function

f(x) - f(2x)

has the derivative

5

at

x = 1

and derivative

7

at

x = 2

. The derivative of the function

f(x) - f(4x) - 10x

at

x = 1

is equal to.

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medium

If

f(x) = x^3 + x^2 f'(1) + x f''(2) - f'''(3)

for all

x \in \mathbb{R}

, then

|f(0)+f(3)+16|

is.

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medium

Let

f:\mathbb{R}\to\mathbb{R}

and

g:\mathbb{R}\to\mathbb{R}

be twice differentiable functions satisfying

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

,

2f'(1) = g'(1) = 4

, and

3f(2) = g(2) = 9

. Then the value of

15+f(4)-g(4)

is equal to.

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medium

If

y = \tan^{-1}\dfrac{4x}{1+5x^2}+\tan^{-1}\dfrac{2+3x}{3-2x}

and

\dfrac{dy}{dx} = \dfrac{k}{1+25x^2}

, then

k

is.

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medium

If

y = e^{\sin^{-1}x}+e^{-\cos^{-1}x}

, then the value of

\left|\dfrac{y''(x^2-1)+y}{xy'}\right|

.

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medium

A function

f:\mathbb{R}\to\mathbb{R}

satisfies

\sin x\cos y\,(f(2x+2y)-f(2x-2y)) = \cos x\sin y\,(f(2x+2y)+f(2x-2y))

. If

f'(0) = \dfrac{1}{2}

, then the value of

4f''(x)+f(x)

is.

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medium

Let a differentiable function

f

satisfy

f(xy) = 2f(x)+f(y)+x+2y+1

for all

x, y \in \mathbb{R}

. If

f'(1) = 1

, then

f(e^3)

is equal to.

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medium

If

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

satisfies

Axy''+By'+Cy = 0

, then

A+B+C

is equal to.

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medium

If

f(x) = \lambda x^2+\mu x+12

,

f'(4) = 15

and

f'(2) = 11

, then

\lambda+\mu

is.

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medium

If

f(x) = (ax+b)\sin x+(cx+d)\cos x

, then the values of

a, b, c, d

such that

f'(x) = x\cos x

for all

x

, then

(a+b+c+d)

is.

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medium

y = \dfrac{2x^2}{(x-2)(x-3)(x-4)} + \dfrac{3x}{(x-3)(x-4)} + \dfrac{x}{x-4}

. If

\dfrac{x}{y}\dfrac{dy}{dx} = \dfrac{a}{2-x}+\dfrac{b}{3-x}+\dfrac{c}{4-x}

, then

a+b+c

is.

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medium

If

x^2 + y^2 = t - \dfrac{1}{t}

and

x^4 + y^4 = t^2 + \dfrac{1}{t^2}

, then

x^3 y\dfrac{dy}{dx}

is.

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easy

If

y=\dfrac{1+x^{2}+x^{4}}{1+x+x^{2}}

and

\dfrac{dy}{dx}=ax+b

, then the values of

a

and

b

are

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easy

If

f(x) = |\cos x|

, then

f'\left(\frac{3\pi}{4}\right)

is:

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easy

If

f(x)=\log x

, then

f'(\log x)

equals

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easy

\dfrac{d}{dx}\left[\dfrac{\sin x + \cos x}{\sqrt{1 + \sin 2x}}\right]

, where

0 < x < \dfrac{\pi}{4}

, is:

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easy

The second derivative of a single valued function parametrically represented by

x = \phi(t)

and

y = \psi(t)

, where

\alpha < t < \beta

and

\phi(t)

,

\psi(t)

are differentiable functions, is given by:

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easy

If

\dfrac{d}{dx}\left\{\dfrac{x}{2}\sqrt{a^2 - x^2} + \dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a}\right\} = f(x)

, then

y = (f(x))^2 + x^2

is:

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easy

If

y

is a function of

x

and

\ln(x+y) = 2xy

, then find the value of

y''(0)

.

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easy

If

y

is a function of

x

and

\log(x+y) = 2xy

, then

y'(0)

is.

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easy

If

f'(x) = \sqrt{3x^2+6}

and

y = f(x^3)

, then at

x = 1

,

\left(\dfrac{dy}{dx}\right)

is equal to.

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Method of Differentiation — JEE Maths Practice Questions & Solutions

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