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

SectION 14.3 Partial Derivatives 917
so
Now we regard m as a constant. The partial derivative with respect to h is
−B
−h
−B
−h
−
sm, hd − S 2D 3D m − mS2 2 − 2 2m
−h h h h 3
s64, 1.68d − 2
2 ? 64
s1.68d 3 < 227 skgym2 dym
This is the rate at which the man’s BMI increases with respect to his height when he
weighs 64 kg and his height is 1.68 m. So if the man is still growing and his weight
stays unchanged while his height increases by a small amount, say 1 cm, then his BMI
will decrease by about 27s0.01d − 0.27.
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EXAMPLE 4 If f sx, yd − sinS x calculate
1 1 yD, −f −f
and
−x −y .
SOLUTION Using the Chain Rule for functions of one variable, we have
−f
S
−x − cos x ?
1 1 yD
−f
S
−y − cos x ?
1 1 yD
−
S x −
−x 1 1 yD cosS x ?
1 1 yD
−
S x −
−y 1 1 yD 2cosS x ?
1 1 yD
1
1 1 y
x
s1 1 yd 2
■
Some computer software can plot
surfaces defined by implicit equations
in three variables. Figure 6 shows such
a plot of the surface defined by the
equation in Example 5.
EXAMPLE 5 Find −zy−x and −zy−y if z is defined implicitly as a function of x and y by
the equation
x 3 1 y 3 1 z 3 1 6xyz − 1
SOLUTION To find −zy−x, we differentiate implicitly with respect to x, being careful to
treat y as a constant:
3x 2 1 3z −z
−z
2 1 6yz 1 6xy
−x −x − 0
Solving this equation for −zy−x, we obtain
−z
−x − 2 x 2 1 2yz
z 2 1 2xy
FIGURE 6
Similarly, implicit differentiation with respect to y gives
−z
−y − 2 y 2 1 2xz
z 2 1 2xy
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Functions of More Than Two Variables
Partial derivatives can also be defined for functions of three or more variables. For
example, if f is a function of three variables x, y, and z, then its partial derivative with
respect to x is defined as
f x sx, y, zd − lim
h l 0
f sx 1 h, y, zd 2 f sx, y, zd
h
and it is found by regarding y and z as constants and differentiating f sx, y, zd with respect
to x. If w − f sx, y, zd, then f x − −wy−x can be interpreted as the rate of change of w with
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