Derivative of ln

$y = l n (x)$

$e^{y} = e^{l n (x)}$

Note: $e^{l n (x)} = x$
$e^{y} = x$

Implicit differentiation

$\frac{d}{d x} e^{y} = \frac{d x}{d x}$

$\frac{d y}{d x} \cdot e^{y} = 1$

$\frac{d y}{d x} = \frac{1}{e^{y}}$

Recall above: $e^{y} = x$

$\frac{d y}{d x} = \frac{1}{x}$