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

212 Chapter 3 Differentiation Rules
2
(Note that we have used the Chain Rule on the left side and the Product Rule and Chain
Rule on the right side.) If we collect the terms that involve y9, we get
cossx 1 yd 1 y 2 sin x − s2y cos xdy9 2 cossx 1 yd ? y9
_2 2
So
y9 − y 2 sin x 1 cossx 1 yd
2y cos x 2 cossx 1 yd
FIGURE 6
_2
Figure 6, drawn with the implicit-plotting command of a computer algebra system,
shows part of the curve sinsx 1 yd − y 2 cos x. As a check on our calculation, notice
that y9 − 21 when x − y − 0 and it appears from the graph that the slope is approximately
21 at the origin.
■
Figures 7, 8, and 9 show three more curves produced by a computer algebra system
with an implicit-plotting command. In Exercises 41–42 you will have an opportunity to
create and examine unusual curves of this nature.
4
15
12
_4 4
_15 15
_12 12
_4
FIGURE 7
sx 2 2 1dsx 2 2 4dsx 2 2 9d
− y 2 sy 2 2 4dsy 2 2 9d
_15
FIGURE 8
cossx 2 sin yd − sinsy 2 sin xd
_12
FIGURE 9
sinsxyd − sin x 1 sin y
The following example shows how to find the second derivative of a function that is
defined implicitly.
ExamplE 4 Find y99 if x 4 1 y 4 − 16.
SOLUTION Differentiating the equation implicitly with respect to x, we get
Solving for y9 gives
4x 3 1 4y 3 y9 − 0
3 y9 − 2 x 3
To find y99 we differentiate this expression for y9 using the Quotient Rule and remembering
that y is a function of x:
y99 − d dx
S2 x 3
y 3
3D − 2 y 3 sdydxdsx 3 d 2 x 3 sdydxdsy 3 d
y sy 3 d 2
− 2 y 3 ? 3x 2 2 x 3 s3y 2 y9d
y 6
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