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If f(x) is an odd, twice derivable function such that f(–1) = f(2) = f(3) = –1, then minimum number of roots of the equation (f′(x))² + f′′(x) = 0 is

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Important Questions on Differential Coefficient

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Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

[x]

represents the greatest integer function and

f (x) = x - [x] - \cos x

then

f^{'} (\frac{π}{2}) =

MEDIUM

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

y (α) = \sqrt{2 (\frac{t a n α + c o t α}{1 + t a n^{2} α}) + \frac{1}{s i n^{2} α}}, α \in (\frac{3 π}{4}, π)

, then

\frac{d y}{d α}

α = \frac{5 π}{6}

MEDIUM

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

f (x) = \sin^{- 1} [\frac{2^{x + 1}}{1 + 4^{x}}],

then

f^{'} (0) =

EASY

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

f (x) = \log_{x^{2}} (\log x),

then

f^{'} (e)

is equal to __________

HARD

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

Let

f

be a polynomial function such that

f (3 x) = f^{'} (x) . f^{''} (x),

for all

x \in R .

Then :

EASY

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

Let

f (x) = x [x], x \notin Z

, (

[\cdot]

denotes greatest integer function), then

f^{'} (x)

is equal to

MEDIUM

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

Let

f (x) = {(\sin (\tan^{- 1} x) + \sin (\cot^{- 1} x))}^{2} - 1

|x| > 1

. If

\frac{d y}{d x} = \frac{1}{2} \frac{d}{d x} (\sin^{- 1} (f (x)))

and

y (\sqrt{3}) = \frac{π}{6},

then

y (- \sqrt{3})

is equal to:

EASY

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

{(x e)}^{y} = e^{x}

, then

\frac{d y}{d x}

HARD

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

Let

p (x)

be a polynomial such that

p (x) - p^{'} (x) = x^{n}

where

n

is a positive integer. Then,

p (0)

equals

EASY

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

f (1) = 1, f^{'} (1) = 3

, then the derivative of

f (f (f (x))) + {(f (x))}^{2}

x = 1

is:

MEDIUM

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

\frac{d}{d x} (\frac{x^{4} + x^{2} + 1}{x^{2} + x + 1}) = a x + b,

then

a - b =

EASY

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

If the function

f (x)

defined by

f (x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . . . . + \frac{x^{2}}{2} + x + 1,

then

f^{'} (0) =

HARD

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

y = \tan^{- 1} (\frac{4 x}{1 + 5 x^{2}}) + \tan^{- 1} (\frac{2 + 3 x}{3 - 2 x}),

then

\frac{d y}{d x} =

HARD

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

y = \sum_{k = 1}^{6} k \cos^{- 1} \{\frac{3}{5} \cos k x - \frac{4}{5} \sin k x\}

, then

\frac{d y}{d x}

x = 0

MEDIUM

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

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

, then

f^{'} (\sqrt{3})

MEDIUM

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

The derivative of

\cos h^{- 1} x

with respect to

\log x

x = 5

HARD

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

Suppose,

f (x) = e^{- \sqrt{x}} + e^{- \frac{1}{x^{2}}}

. If

f^{''} (x) = α \cdot \frac{e^{- \sqrt{x}}}{x} (1 + \frac{1}{\sqrt{x}}) + β \cdot \frac{e^{- \frac{1}{x^{2}}}}{x^{4}} (3 - \frac{2}{x^{2}})

, then

(α, β) =

MEDIUM

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

Let

x^{k} + y^{k} = a^{k}, (a, k > 0)

and

\frac{d y}{d x} + {(\frac{y}{x})}^{\frac{1}{3}} = 0,

then

k

MEDIUM

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

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

then

y^{'}

is equal to

MEDIUM

Mathematics>Differential Calculus>Differential Coefficient>Basics of Differentiation

y = \frac{x + c}{1 + x^{2}}

where

c

is a constant, when

y

is stationary,

x y

is equal to -

If f(x) is an odd, twice derivable function such that f(–1) = f(2) = f(3) = –1, then minimum number of roots of the equation (f′(x))2 + f′′(x) = 0 is

Important Questions on Differential Coefficient

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If f(x) is an odd, twice derivable function such that f(–1) = f(2) = f(3) = –1, then minimum number of roots of the equation (f′(x))² + f′′(x) = 0 is