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2.6. DERIVATIVES OF INVERSE FUNCTIONS
g(x) = ln(x) and f (x) = e x , it follows that
g
′ (x) =
1
f ′ (g(x))
=
1
e ln(x) =
1
x
.
Summary
In this section, we encountered the following important ideas:
• For all positive real numbers x,
d
dx
[ln(x)] =
1
x
.
• For all real numbers x such that −1 ≤ x ≤ 1,
d
dx
[arcsin(x)] =
1
√
1 − x 2
. In addition,
for all real numbers x,
d
dx
[arctan(x)] =
1
1 + x 2 .
• If g is the inverse of a differentiable function f , then for any point x in the domain of
g ′ ,
g
′ (x) =
1
f ′ (g(x))
.
Exercises
1. Determine the derivative of each of the following functions. Use proper notation and
clearly identify the derivative rules you use.
(a) f (x) = ln(2 arctan(x) + 3 arcsin(x) + 5)
(b) r(z) = arctan(ln(arcsin(z)))
(c) q(t) = arctan 2 (3t) arcsin
4 (7t)
(d) g(v) = ln
arctan(v)
arcsin(v) + v 2
2. Consider the graph of y = f (x) provided in Figure 2.13 and use it to answer the
following questions.
(a) Use the provided graph to estimate the value of f ′ (1).
(b) Sketch an approximate graph of y = f −1 (x). Label at least three distinct points
on the graph that correspond to three points on the graph of f .
(c) Based on your work in (a), what is the value of ( f −1 ) ′ (−1)? Why?
3. Let f (x) =
1
4 x 3 + 4.
(a) Sketch a graph of y = f (x) and explain why f is an invertible function.