## Differentiation of Exponential Functions

The exponential function f(x) = ex is a special function. It is the only function which is its own derivative.

Some people write ex as exp(x).

e is an irrational number approximately equal to 2.718282 (to 6 d.p.)

This means that at any point on the graph of the exponential function the gradient of the curve at that point is equal to the value of the function (the y-value) at that point.

The function can also be written as a series.

It can be seen that if this series is differentiated term by term the result is the same series!

If x = 1 then the value of e can be worked out to the required accuracy using the series.

This means that:

 If f(x) = ex then f '(x) = ex

### Using the Chain Rule with Exponential Functions

To differentiate functions such as f(x) = e2x the chain rule is used.

example f(x) = e2x

f '(x) = 2. e2x = 2e2x

In general terms:

 If f(x) = eax then f '(x) = aeax

example f(x) = 3e200x

f '(x) =3e200x . 200

= 600e200x