James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

214 Chapter 3 Differentiation Rules
The same method can be used to find
a formula for the derivative of any
inverse function. See Exercise 77.
Therefore
dy
dx − 1
cos y − 1
s1 2 x 2
d
dx ssin21 xd −
1
s1 2 x 2
Figure 11 shows the graph of
f sxd − tan 21 x and its derivative
f 9sxd − 1ys1 1 x 2 d. Notice that f is
increasing and f 9sxd is always positive.
The fact that tan 21 x l 6y2 as
x l 6` is reflected in the fact that
f 9sxd l 0 as x l 6`.
1.5
y=tan–! x
y= 1
1+≈
_6 6
The formula for the derivative of the arctangent function is derived in a similar way.
If y − tan 21 x, then tan y − x. Differentiating this latter equation implicitly with respect
to x, we have
sec 2 y dy
dx − 1
dy
dx − 1
sec 2 y − 1
1 1 tan 2 y − 1
1 1 x 2
d
dx stan21 xd − 1
1 1 x 2
FIGURE 11
_1.5
ExamplE 5 Differentiate (a) y − 1
sin 21 x and (b) f sxd − x arctansx .
SOLUTION
dy
(a)
dx − d
dx ssin21 xd 21 − 2ssin 21 xd d 22 dx ssin21 xd
1
− 2
ssin 21 xd 2 s1 2 x 2
Recall that arctan x is an alternative
notation for tan 21 x.
(b)
f 9sxd − x
1
1 1 (sx ) 2 ( 1 2 x21y2 ) 1 arctansx
−
sx
2s1 1 xd 1 arctansx
■
The inverse trigonometric functions that occur most frequently are the ones that we
have just discussed. The derivatives of the remaining four are given in the following
table. The proofs of the formulas are left as exercises.
The formulas for the derivatives of
csc 21 x and sec 21 x depend on the
definitions that are used for these
functions. See Exercise 64.
Derivatives of Inverse Trigonometric Functions
d
dx ssin21 xd −
1
s1 2 x 2
d
1
dx scos21 xd − 2
s1 2 x 2
d
dx stan21 xd − 1
1 1 x 2
d
dx scsc2 1
1
xd − 2
xsx 2 2 1
d
dx ssec2 1
xd −
1
xsx 2 2 1
d
dx scot2 1
xd − 2 1
1 1 x 2
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