Definition of the derivative

For a function f(x), its derivative f`(x) is defined as follows:

In simple words the above definition is saying that the derivative of a function is another function that gives the instantaneous rate of change of that function.

The derivative of a function gives the rate of change of a function at any point on it's graph. Limits are used in the definition of the derivative because the limit considers the tiniest region around a particular point on the function and gives its average value in that tiniest region.

Using the definition of the derivative to evaluate the defrivative of simple functions:

The above definition can be applied to find the actual derivative of functions. For example, let
f(x) = x^2 + 2x + 1. Let its derivative be f`(x). By the above definition, f`(x) is given by
Now evaluate the limit by direct substitution:
= 2x + 2
So the derivative of f(x) = x^2 + 2x + 1 is equal to 2x + 2