# The derivative of ln(x)

## Statement

The derivative of $\ln(x)$ is $\frac{1}{x}$:

$$\tfrac{d}{dx}\bigg(\ln(x)\bigg) = \frac{1}{x}$$

## Proof

Let $y = \ln(x)$.

Raise both sides to a power with base $e$. Since $e^{...}$ and $\ln(...)$ are inverse functions, they cancel.

$$e^y = e^{\ln(x)}$$

$$e^y = x$$

Differentiate both sides with respect to $x$. The left sides is differentiated using the derivative of $e^x$ and the chain rule. The right side is just $1$, thanks to the power rule.

$$\tfrac{d}{dx}(e^y) = \tfrac{d}{dx}(x)$$

$$e^y * \frac{dy}{dx} = 1$$

Finally, divide both sides by $e^y$ and substitute $e^y = x$.

$$\frac{dy}{dx} = \frac{1}{e^y} = \frac{1}{x}$$

## Proofs building upon this proof

### Log rule in calculus

This proofs shows the derivative of a logarithmic function.