As we noticed in the geometrical interpretation of differentiation, we can find the derivative of a function at a given point. If the derivative exists for every point of the function, then it is defined as the derivative of the function f(x).

Suppose f(x) is a real valued function, the function defined by

For this function, if the derivative exists at every point along the curve, then we say

f'(x) =

This definition of derivative of f(x) is called the **First Principle of Derivatives**.

The function f'(x) or is called the gradient function.

The process of finding the gradient value of a function at any point on the curve is called differentiation, and the gradient function is called the derivative of f(x).

There are different ways of representing the derivative of a function:

, , f'(x), y’, , and

**Example 1**: Find the derivative of f(x) = 5x using first principles.

f(x + h) = 5(x + h)

=

=

= 5

f'(x) = = 5

Therefore f(x) = 5x; and f'(x) = 5.

**Example 2**: Find the derivative of f(x) = 5x^{2} + 2x.

f(x + h) = 5(x + h)^{2} + 2(x + h)

= 5(x^{2} + 2xh + h^{2}) + 2x + 2h

= 5x^{2} + 10xh + 5h^{2} + 2x + 2h

f(x + h) – f(x) = (5x^{2} + 10xh + 5h^{2} + 2x + 2h) – (5x^{2} + 2x)

= 10xh + 5h^{2} + 2h

f'(x) =

=

=

= 10x + 2

There are more examples of derivatives here.