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 xn-1

Derivative of constant:

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

If x = x (where x0 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 x0 with respect to t

`dx_0/dt`   which can also be written `d/dt` (x0)

`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 = x0 (where xo is a constant) then v = 0. x = x0 .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 problems:

 

Derivative problem 1:

Find the derivative of  equation y = x4 + 3x2 - 84 with respect to x,

Solution:

Given equation, y = x4 + 3x2 - 84

Differentiate with respect to x .

` dy/dx` = `d/dx` ( x4 + 3x2 - 84)

= `d/dx ` (x4 ) + 3`d/dx` (x2 ) - `d/dx` (84)                            we know, ` d / dx` (x n ) = n xn-1

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

= 4 x3 + 6x - 0

= 4 x3 + 6x

Answer: `dy/dx` = 4 x3 + 6x

 

Derivative problem 2:

Find the derivative of  equation y = 4x4 - 2x3 - 4 with respect to x,

Solution:

Given equation, y = 4x4 - 2x3 - 4

Differentiate with respect to x .

` dy/dx` = `d/dx` ( 4x4 - 2x3 - 4)

= 4`d/dx ` (x4 ) - 2`d/dx` (x3 ) - `d/dx` (4)                            we know, ` d / dx` (x n ) = n xn-1

= (4 × 4) x(4 - 1) - (2 × 3) x(3 - 1) - 4(0)

= 16 x3 - 6x2 - 0

= 16 x3 - 6x2

Answer: `dy/dx` =16 x3 - 6x2