5128_Ch03_pp098-184

168 Chapter 3 Derivatives
–1
y
y sec –1 x
Domain: |x| 1
Range: [0, /2) ∪ (/2, ]
–2
x
0 1
≤
To express the result in terms of x, we use the relationships
to get
sec y x and tan y sec 2 y 1 x 2 1
dy 1
.
d x xx 2 1
Figure 3.54 The slope of the curve
y sec 1 x is positive for both x 1
and x 1.
Can we do anything about the sign? A glance at Figure 3.54 shows that the slope of the
graph y sec 1 x is always positive. That must mean that
1
{
if x 1
d
sec d x
1 xx2 1
x
1
if x 1.
xx2 1
With the absolute value symbol we can write a single expression that eliminates the
“” ambiguity:
d
sec d x
1 1
x .
xx 2 1
If u is a differentiable function of x with u 1, we have the formula
d
sec d x
1 1
u d u
, u 1.
uu 2 1 dx
EXAMPLE 3
Using the Formula
d
sec d x
1 5x 4 1 d
5x 5x 4 5 x 4
2 1 d x
4
1
20x 3
5x
4
2 5x 8 1
4
x25 x 8 1
Now try Exercise 17.
Derivatives of the Other Three
We could use the same technique to find the derivatives of the other three inverse trigonometric
functions: arccosine, arccotangent, and arccosecant, but there is a much easier way,
thanks to the following identities.
Inverse Function–Inverse Cofunction Identities
cos 1 x p2 sin 1 x
cot 1 x p2 tan 1 x
csc 1 x p2 sec 1 x