Differentiating

Remember:

1. *e* is a number whose value is about 2.72.

2. *e* is used as the base for the exponential and logarithm functions,
*e*^{x} and .

The function *e*^{x} has the following amazing property:

The derivative of *e*^{x} is just *e*^{x} again. That's kind of like giving birth
to yourself, a very difficult event to achieve, and one that would get you
major tabloid coverage.

**NEWS FLASH: ``I AM MY OWN MOTHER," FUNCTION CLAIMS!**

In fact, *e*^{x} and its multiples are the only functions that are their
own derivatives. For this reason we put up with the use of the very strange number
2.718281828459... as the base.

Notice that
is the composition of *e*^{x} with .
So we'll use the chain rule to differentiate it.

The
is the derivative of the outside function *e*^{u}evaluated at the inside function
and the
is the derivative
of the inside function.