# Derivative of *a*^{x}

How to solve the derivative of *a*^{x} problems: formula, proof, example, and its solution.

## Formula

The derivative of *a*^{x} is *a*^{x} ln *a*.

## Proof

ln both sides.

Natural logarithms

Differentiate both sides.

Implicit differentiation

Product rule in differentiation

Multiply *y* on both sides.

Change *y* into *a*^{x}.

Then *y*' = *a*^{x} ln |*a*|.

## Example

*y* = 2^{2x - 1}

So *y*' is equal to,

the derivative of the outer part, (2^{2x - 1}) ln 2

times, the derivative of the inner part, 2.

Chain rule in differentiation

Derivative of polynomials