Derivatives of Inverse Functions

Suppose that we know all about a function f and its derivative f ′. If f has an inverse, g, can we use our knowledge of f to compute the derivative of g? Yes!

If f and g are inverse functions, then

g′(x) =

f ′(g(x))

In the applet above, we will see a geometric justification for this formula. Drag the slider through the steps and consider the questions below.

Explore

Step 0. We are given f and g and our goal is to compute g′(c). Just by looking at the graphs, what tells you that f and g are inverses? . Can you express d in terms of g and c?
Step 1. Justify: if point (c, d) lies on the graph of g, then the point (d, c) lies on the graph of f .
Step 2. Consider the line tangent to f through (d, c). In the applet, the slope of the line is labeled as m. What is m in terms of f , c, and d?
Step 3. Now consider the line tangent to g through (c, d). Remember that our goal was to compute g′(c), in other words, the slope of this line. What is the relation between the slope of this line and the slope of the line in step 2? Move the red point on the x axis to help you see this.
Finally, justify the boxed formula in the introduction.

Alternate Proof

If f and g are inverse functions and x is in the domain of g, then

f (g(x)) = x

Take the derivative of both sides, using the chain rule on the left:

f ′(g(x))g′(x) = 1

Solve for g′(x):

g′(x) =

f ′(g(x))

Done!