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
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)
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
= 10x + 2
There are more examples of derivatives here.