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) = 5x2 + 2x.
f(x + h) = 5(x + h)2 + 2(x + h)
= 5(x2 + 2xh + h2) + 2x + 2h
= 5x2 + 10xh + 5h2 + 2x + 2h
f(x + h) – f(x) = (5x2 + 10xh + 5h2 + 2x + 2h) – (5x2 + 2x)
= 10xh + 5h2 + 2h
f'(x) =
= 
= 
= 10x + 2
There are more examples of derivatives here.