108 Chapter 4Differentiation
Example 4.15(i) demonstrates the important special case of u(x) 1 = 1 ax, for which
. Then
For example,
Inverse functions
Ify 1 = 1 f(x), the inverse function offis defined byx 1 = 1 f
− 1(y). By Rule 6 in Table 4.3,
the derivatives of function and inverse function are related by
The inverse rule
(4.14)
is used when it is more difficult to differentiate the function than its inverse.
EXAMPLE 4.16Use of the inverse rule
If yis defined implicitly by
x 1 = 1 y
51 − 12 y
(see Example 2.11), then can be found as the inverse of :
0 Exercises 56–59
Particularly important examples of the differentiation of inverse functions are
given in Table 4.5, where the inverse trigonometric functions have their principal
values.
dx
dy
y
dy
dx
y
=−, =
−
52
1
52
44dx
dy
dy
dx
dx
dy
dydx
=
1
dy
dx
dx
dy
d
dx
fx
d
dy
×= fy
×
−() ()
1== 1
d
dx
xx
d
dx
ee
xxcos sin 333 2
22=− , =
dy
dx
dy
du
du
dx
a
dy
du
=×=
du
dx
=a