92 STEP 4. Review the Knowledge You Need to Score High
[−2,4] by [−2,4]
Figure 6.4-2
TIP • Remember that the lim
x→ 0
sin 6x
sin 2x
=
6
2
=3 because the limx→ 0
sinx
x
=1.
6.5 Derivatives of Inverse Functions
Let f be a one-to-one differentiable function with inverse function f−^1 .If
f′(f−^1 (a))/=0, then the inverse function f−^1 is differentiable ata and (f−^1 )′(a) =
1
f′(f−^1 (a))
. (See Figure 6.5-1.)
f' (f–1(a))
1
(a, f–1(a))
y
x
0
y=x
f
f–1
(f–1(a),a)
m=(f–1)'(a)
m=f'(f–1(a))
(f–1)' (a) =
Figure 6.5-1
Ify=f−^1 (x) so thatx=f(y), then
dy
dx
=
1
dx/dy
with
dx
dy
=/0.
Example 1
Iff(x)=x^3 + 2 x−10, find (f−^1 )′(x).
Step 1: Check if (f−^1 )′(x) exists. f′(x)= 3 x^2 +2 and f′(x) > 0 for all real values
ofx. Thus,f(x) is strictly increasing, which implies thatf(x)is1−1. Therefore,
(f−^1 )′(x) exists.
Step 2: Lety=f(x) and thusy=x^3 + 2 x−10.
Step 3: Interchangexandyto obtain the inverse functionx=y^3 + 2 y−10.
Step 4: Differentiate with respect toy:
dx
dy
= 3 y^2 + 2.
Step 5: Apply formula
dy
dx
=
1
dx/dy
.
dy
dx
=
1
dx/dy
=
1
3 y^2 + 2
.Thus, (f−^1 )′(x)=
1
3 y^2 + 2