Deriving the Power Rule
Let
. Our goal is to find
.
So, first we apply the definition of the derivative
to. So we then have that
.
We expand the expression 
to obtain
We only need to be careful about expanding the expression on the first two terms, because, as we shall see in a moment, the remaining terms will not play a significant role.
Now, we continue determining the derivative
,
which can be simplified to
and further simplified to
.
If we take the limit as 
, we obtain our result 
as the remaining terms go to zero.