Derivative Proof of a^x

This proof is similar to e^x. In fact, e can be plugged in for a, and we would get the same answer because ln(e) = 1.

Derivative proof of a^x

Rewrite a^x as an exponent of e^ln

Use Chain Rule and substitute u for xlna

Cancel out dx over dx, and substitute back in for u

Bring down the a^x

Here is a graph of f(x) = .5^x and f(x) = 2^x and their derivatives.