Definitions

We define the function exp(x) by

exp(0)= 1

and

These two facts imply

 (the derivative of each term is its predecessor)

exp(1) = e = 2.71828

Graph

Properties

1. exp(0) = 1, exp(1) = e

2. exp(u + v) = exp(u)exp(v)

3. exp(uv) = (exp(u))^v

4.

Applying the third property to u = 1 and using the first, we get exp(v) = e^v.

From now on, we will use the power notation expressed in this formula for the exponent function.

Proof of 2

Proof of 3

Proof of 4

Derivative

(by definition.)