## 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.

$$ f'(x) = a^x * \ln(a) $$