2.6. DERIVATIVES OF INVERSE FUNCTIONS
139
y = f (x)
Figure 2.13: A function y = f (x) for use in Exercise 2.
(b) Let g be the inverse of f and determine a formula for g.
(c) Compute f ′ (x), g ′ (x), f ′ (2), and g ′ (6). What is the special relationship between
f ′ (2) and g ′ (6)? Why?
4. Let h(x) = x + sin(x).
(a) Sketch a graph of y = h(x) and explain why h must be invertible.
(b) Explain why it does not appear to be algebraically possible to determine a
formula for h −1 .
(c) Observe that the point (
π
2 ,
π
2 + 1) lies on the graph of y = h(x). Determine the
value of (h −1 ) ′ (
π
2 + 1).
139
y = f (x)
Figure 2.13: A function y = f (x) for use in Exercise 2.
(b) Let g be the inverse of f and determine a formula for g.
(c) Compute f ′ (x), g ′ (x), f ′ (2), and g ′ (6). What is the special relationship between
f ′ (2) and g ′ (6)? Why?
4. Let h(x) = x + sin(x).
(a) Sketch a graph of y = h(x) and explain why h must be invertible.
(b) Explain why it does not appear to be algebraically possible to determine a
formula for h −1 .
(c) Observe that the point (
π
2 ,
π
2 + 1) lies on the graph of y = h(x). Determine the
value of (h −1 ) ′ (
π
2 + 1).