H. DERIVATIVE OF THE INVERSE OF A FUNCTION
Suppose f and g are inverse functions. What is the relationship between their
derivatives? Recall that the graphs of inverse functions are the reflections of
each other in the line y = x, and that at corresponding points their x- and y-
coordinates are interchanged.
Figure N3–8 shows a function f passing through point (a,b) and the line
tangent to f at that point. The slope of the curve there, f ′(a), is represented by the
ratio of the legs of the triangle, . When this figure is reflected across the line y
= x, we obtain the graph of f−^1 , passing through point (b,a), with the horizontal
and vertical sides of the slope triangle interchanged. Note that the slope of the
line tangent to the graph of f−^1 at x = b is represented by , the reciprocal of the
slope of f at x = a. We have, therefore,