Derivative of a^x/Proof

$\frac{d}{dx}(a^x) = \ln(a) \cdot a^x$

Proof

Proof using other derivative formulas:

Since the logarithm is the inverse of the exponential, applying logarithm power rules we get

$\begin{align} \frac{d}{dx}(a^x) & = \frac{d}{dx}(e^{\ln(a^x)}) \\ & = \frac{d}{dx}(e^{x\ln(a)}). \end{align}$

Applying the chain rule to the above and noting that the derivative of $x$ is $1$, we get the result.

$\begin{align} &= e^{x \ln(a)}\frac{d}{dx}\left(x \ln(a)\right)\\ &= e^{x \ln(a)}\ln(a)\frac{d}{dx}\left(x\right)\\ &= e^{x \ln(a)}\ln(a)\\ &= \ln(a)a^{x}. \end{align}$