## 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 $.

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