# Derivative of General Exponential Function

## Theorem

Let $a \in \R: a > 0$.

Let $a^x$ be $a$ to the power of $x$.

Then:

$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$

## Proof 1

 $\ds \map {\frac \d {\d x} } {a^x}$ $=$ $\ds \map {\frac \d {\d x} } {e^{x \ln a} }$ Definition of Power to Real Number $\ds$ $=$ $\ds \ln a e^{x \ln a}$ Derivative of Exponential of a x $\ds$ $=$ $\ds a^x \ln a$

$\blacksquare$

## Proof 2

 $\ds \lim_{h \mathop \to 0} \frac {a^{x + h} - a^x} h$ $=$ $\ds a^x \lim_{h \mathop \to 0} \frac {a^h - 1} h$ Product of Powers $\ds$ $=$ $\ds a^x \lim_{h \mathop \to 0} \frac {\map \exp {h \ln a} - 1} h$ Definition of Power to Real Number $\ds$ $=$ $\ds a^x \lim_{h \mathop \to 0} \paren {\frac {\map \exp {h \ln a} - 1} {h \ln a} } \paren {\frac {h \ln a} h}$ $\ds$ $=$ $\ds a^x \lim_{h \mathop \to 0} \paren {\frac {h \ln a} h}$ Derivative of Exponential at Zero $\ds$ $=$ $\ds a^x \ln a$

$\blacksquare$

## Proof 3

Let $y = a^x$.

Then:

 $\ds \ln y$ $=$ $\ds x \ln a$ Logarithm of Power $\ds \leadsto \ \$ $\ds \dfrac 1 y \dfrac {\d y} {\d x}$ $=$ $\ds \ln a$ Derivative of Identity Function: Corollary $\ds \leadsto \ \$ $\ds \dfrac 1 {a^x} \dfrac {\d y} {\d x}$ $=$ $\ds \ln a$ $\ds \leadsto \ \$ $\ds \dfrac {\d y} {\d x}$ $=$ $\ds a^x \ln a$

$\blacksquare$