Statement
The derivative of $ a^x $ is $ a^x * \ln(a) $:
$$ \tfrac{d}{dx}(a^x) = a^x * \ln(a) $$
Proof
Define function $ f $.
$$ f(x) = a^x $$
Now write it using $ e^{...} $ and $ \ln(...) $. This is allowed because they are inverse functions.
$$ f(x) = e^{\ln(a^x)} $$
Differentiate this function using the derivative of $ e^x $ and the chain rule.
$$ f'(x) = e^{\ln(a^x)} * \tfrac{d}{dx}\bigg(\ln(a^x)\bigg) $$
For the first factor, $ e^{...} $ and $ \ln(...) $ cancel because they are inverse functions.
For the second factor, multiply the logarithm by the exponent because of the power rule.
$$ f'(x) = a^x * \tfrac{d}{dx}\bigg(\ln(a) * x\bigg) $$
Finally, differentiate the final derivative using the coefficient rule and the power rule.
$$ f'(x) = a^x * \ln(a) $$