The differential of y if y = x 4 + x 2 + 1 is

1 2 x 4 + x 2 + 1 - 1 2 4 x 3 + 2 x d x

1 2 x 4 + x 2 + 1 - 1 2 4 x 3 + 2 x

MEDIUM

Earn 100

What is the derivative of $\sin x ?$

Important Questions on Limits and Derivatives

EASY

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

Find $\frac{d y}{d x}$ , where $y = \tan^{- 1} x + \cot^{- 1} x$ .

EASY

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

Let

f (x + y) = f (x) f (y)

for all

x

and

y .

f (4) = 3

and

f^{'} (0) = 1,

then

f^{'} (4)

equals.

EASY

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

The differential of $y$ if $y = \sqrt{x^{4} + x^{2} + 1}$ is

EASY

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

Y = x (x - 1) (x - 2)

then

d y / d x

MEDIUM

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

Let $K$ be a set of real number and $f : K \to R$ such that for all $x$ and $y,$ $|f (x) - f (y)| \leq {|x - y|}^{5} .$

If $f (3) = 7,$ then the value of $f (9)$ is

MEDIUM

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

Let

f : ℝ \to ℝ

be a nonconstant differentiable function such that

\lim_{x \to \infty} f (x) = 1

and

\lim_{x \to \infty} f^{'} (x) = α

for some

α \in R

. Then

EASY

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

How many functions

f : R \to R

satisfy

f (1) = 10

and

|f (x) - f (y)| = |x - y|

for all

x, y \in R ?

MEDIUM

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

Define derivative of a function

f (x)

at the point

x = 0

MEDIUM

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

f (x + y) = f (x) f (y)

for all

x

and

y

and

f (5) = 2, f^{'} (0) = 3

, then

f^{'} (5)

MEDIUM

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

S^{2} = a t^{2} + 2 bt + c

, then the acceleration is

HARD

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

Let

c, k \in R

. If

f (x) = (c + 1) x^{2} + (1 - c^{2}) x + 2 k

and

f (x + y) = f (x) + f (y) - x y

, for all

x, y \in R

, then the value of

|2 (f (1) + f (2) + f (3) + \dots \dots + f (20))|

is equal to ______.

HARD

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

a \neq b, x \neq n π, n \in Z

and

y^{2} = a^{2} \cos^{2} x + b^{2} \sin^{2} x,

then

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

EASY

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

Let

f : ℝ \to ℝ

be a differentiable function that satisfies the relation

f (x + y) = f (x) + f (y) - 1, \forall x

y \in ℝ

. If

f^{'} (0) = 2

, then

| f (- 2) |

is equal to

EASY

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

Differentiate

\sin x

by First Principle.

HARD

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

y = \tan x

, then

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

EASY

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

Let

y = f (x) = 2 x^{2} - 3 x + 2

. The differential of

y

when

x

changes from

2

1.99

EASY

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

f

is differentiable at

x = 1

and

f (1) = 0,

\lim_{h \to 0} \frac{1}{h} f (1 + h) = 5,

then

f^{'} (1) =

MEDIUM

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

f

is a real-valued differentiable function satisfying

|f (x) - f (y)| \leq {(x - y)}^{2}, x, y \in R

and

f (0) = 0

, then

f (1)

equals

MEDIUM

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

Differentiate the following functionfrom first principles

e^{\sqrt{\cot x}}

MEDIUM

Mathematics>Differential Calculus>Limits and Derivatives>First Principle of Differentiation

The derivative of

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

with respect to

\cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}})

What is the derivative of sinx?

Important Questions on Limits and Derivatives

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What is the derivative of $\sin x ?$