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

Geometrically, there is a close relationship between the plots of $$ and $$, they are reflections of each other over the line $$:
One may suspect that we can use the fact that $$, to deduce the derivative of $$. 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: