Derivative of the natural logarithmic function

We know that $y=e^x$ and $x=\ln y$ 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 $y=\ln x$, then $\frac{dy}{dx}=\frac{1}{x}$