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