# Rules of Differentiation: The Power Rule

### Erin Horst

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

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