5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 9, 2018 10:9

126 STEP 4. Review the Knowledge You Need to Score High

[–2, 4] by [–2, 4]

Figure 7.4-2

TIP • Remember that the lim

x→ 0

sin 6x

sin 2x

=

6

2

=3 because the limx→ 0

sinx

x

=1.

7.5 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 7.5-1.)

f′ (f–^1 (a))

1

(a, f–^1 (a))

y

0 x

y = x

f

f–^1

(f–^1 (a), a)

m = (f–^1 )′(a)

m = f′(f–^1 (a))

(f–^1 )′^ (a) =

Figure 7.5-1

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 that f(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

.