FIGURE N3–8
Simply put, the derivative of the inverse of a function at a point is the reciprocal of the derivative
of the function at the corresponding point.
EXAMPLE 32
If f (3) = 8 and f ′(3) = 5, what do we know about f −1?
SOLUTION: Since f passes through the point (3,8), f −1 must pass through the point (8,3).
Furthermore, since the graph of f has slope 5 at (3,8), the graph of f −1 must have slope at (8,3).EXAMPLE 33
A function f and its derivative take on the values shown in the table. If g is the inverse of f, find g
′(6).
SOLUTION: To find the slope of g at the point where x = 6, we must look at the point on f
where y = 6, namely, (2,6). Since f ′(2) = g ′(6) = 3.
x f (x) f ′(x)
2 6
6 8EXAMPLE 34
Let y = f (x) = x^3 + x − 2, and let g be the inverse function. Evaluate g ′(0).
SOLUTION: Since To find x when y = 0, we must solve the equation x^3
+ x − 2 = 0. Note by inspection that x = 1, so