### Derivative of the natural logarithmic function

We know that $y={e}^{x}$ and $x=\mathrm{ln}y$ are inverse functions of each other.

Also, we know that $\frac{dy}{dx}={e}^{x}$.

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

Usually we write any function as a function of $x$, so, if we write $y=\mathrm{ln}x$, then $\frac{dy}{dx}=\frac{1}{x}$