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 :
PIC
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: