# Exponent rule in calculus

## 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.

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