Now that we have explored derivatives, we can now progress to the rules of differentiation. These rules are simply formulas that instruct the learner how to compute derivatives depending on a given function.
The most basic rule of differentiation is the power rule. The rule is
Now, consider why this may be true. We can prove the power rule by using the Binomial Theorem. Consider the following proof of the power rule.
If 
, then 
.
Now, to find the derivative, we need to expand 
using the Binomial Theorem. Doing so, we get the following equivalent
statements
To illustrate the power rule, consider the following functions and their respective derivatives. It is suggested that you compute the derivative to verify the function of the derivative.
The following graph illustrates the function 
and its derivative 
The following graph illustrates the function 
and
its derivative 
