** Introduction:**

Calculus is a branch of mathematics. Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x. The derivative of y with respect to x can be denoted as, (dy)/(dx). This functional relationship is often denoted y = ƒ(x), where ƒ denotes the function.

**Source Wikipedia.**

Basic Derivative formula:

` d / dx` (x ^{n} ) = n x^{n-1}

**Derivative of constant:**

The derivative of constant is zero. Zero itself is a constant. so it is no exception.

If x = x_{0 } (where x_{0} is a constant and x represents the position of an object on a straight line path.

the derivative of x with respect to t

` dx/dt ` which can also be written `d/dt` (x)

Then derivative of x_{0} with respect to t

`dx_0/dt` which can also be written `d/dt` (x_{0})

`dx/dt ` = `dx_0/dt `

we know the derivative of constant is zero.

so, `dx/dt` = 0

Differentiation of position with respect to time is called velocity. v = `dx/dt.`

if x = x_{0} (where xo is a constant) then v = 0. x = x_{0} .The object position does not change.

**For example,** Let y = x

The derivative of y with respect to x.

`dy/dx` = `dx/dx` = 1

Now Let y = 5

The derivative of y with respect to x. But here no x variable. 5 is a constant.

So, the derivative of y `dy/dx` = `d/dx` (5) = 0 ( we know zero is a constant)

So, the derivative of zero is zero.

`d/dx` (0) = 0.

**Derivative problem 1:**

Find the derivative of equation y = x^{4} + 3x^{2} - 84 with respect to x,

**Solution:**

Given equation, y = x^{4} + 3x^{2} - 84

Differentiate with respect to x .

` dy/dx` = `d/dx` ( x^{4} + 3x^{2} - 84)

= `d/dx ` (x^{4} ) + 3`d/dx` (x^{2} ) - `d/dx`
(84) we know, `
d / dx` (x ^{n} ) = n x^{n-1}

= 4 x^{(4 - 1)} + (3 × 2) x^{(2 - 1)} - 84(0)

= 4 x^{3} + 6x - 0

= 4 x^{3} + 6x

**Answer:** `dy/dx` = 4 x^{3} + 6x

**Derivative problem 2:**

Find the derivative of equation y = 4x^{4} - 2x^{3} - 4 with respect to x,

**Solution:**

Given equation, y = 4x^{4} - 2x^{3} - 4

Differentiate with respect to x .

` dy/dx` = `d/dx` ( 4x^{4} - 2x^{3} - 4)

= 4`d/dx ` (x^{4} ) - 2`d/dx` (x^{3} ) - `d/dx`
(4) we know, ` d
/ dx` (x ^{n} ) = n x^{n-1}

= (4 × 4) x^{(4 - 1)} - (2 × 3) x^{(3 - 1)} - 4(0)

= 16 x^{3} - 6x^{2} - 0

= 16 x^{3} - 6x^{2}

**Answer:** `dy/dx` =16 x^{3} - 6x^{2}