
We derive the derivatives of inverse exponential functions using implicit differentiation.

Geometrically, there is a close relationship between the plots of $e^x$ and $\ln (x)$, they are reflections of each other over the line $y=x$: One may suspect that we can use the fact that $\ddx e^x = e^x$, to deduce the derivative of $\ln (x)$. We will use implicit differentiation to exploit this relationship computationally.
Compute:

From the derivative of the natural logarithm, we can deduce another fact:

Compute:

We can also compute the derivative of an arbitrary exponential function.

Compute: