Exploring Inverse Functions: From Theory to Real-World Applications

A function is often thought of as a process.  One natural question that arises is, can the process be reversed? If this reversal is possible, we call it the inverse function. Along that same line, when we use the derivative to determine how fast the process is occurring at a particular point, it  raises a very similar question: can we determine how fast the inverse is changing at that same point? 

The Inverse Function Theorem tells us that for an invertible and differentiable function [latex]f(x)[/latex], with [latex]f’(f^{-1}(x)t)(x) \neq 0[/latex], 

[latex]\left(f^{-1}\right)’(x)=\frac{1}{f’\left(f^{-1}(x)\right)}[/latex]

One way to emphasize this reciprocal relationship between the derivative of a function and the derivative of the inverse function is to use something linear.