Derivative of ax

Derivative of a^x

How to solve the derivative of ax problems: formula, proof, example, and its solution.

Formula

[a^x]' = (a^x)*(ln a)

The derivative of ax is ax ln a.

Proof

Derivative of a^x: Proof of the Formula

ln both sides.

Natural logarithms

Logarithms of powers

Differentiate both sides.

Implicit differentiation

Product rule in differentiation

Multiply y on both sides.

Change y into ax.

Then y' = ax ln |a|.

Example

Find the derivative of the given function. y = 2^(2x + 1)

y = 22x - 1

So y' is equal to,
the derivative of the outer part, (22x - 1) ln 2
times, the derivative of the inner part, 2.

Chain rule in differentiation

Derivative of polynomials