SECTION
15.6
The Chain Rule
(ET Section 14.6)
719
Equalities (1) and (2) imply that:
∂
u
∂
t
=−
c
∂
u
∂
x
or
∂
u
∂
t
+
c
∂
u
∂
x
=
0
35.
Jessica and Matthew are running toward the point
P
along the straight paths that make a fxed angle oF
θ
(±igure
3). Suppose that Matthew runs with velocity
v
a
m
/
s and Jessica with velocity
v
b
m
/
s. Let
f
(
x
,
y
)
be the distance From
Matthew to Jessica when Matthew is
x
meters From
P
and Jessica is
y
meters From
P
.
(a)
Show that
f
(
x
,
y
)
=
p
x
2
+
y
2
−
2
xy
cos
.
(b)
Assume that
=
π
/
3. Use the Chain Rule to determine the rate at which the distance between Matthew and Jessica
is changing when
x
=
30,
y
=
20,
v
a
=
4m
/
s, and
v
b
=
3m
/
s.
A
B
x
v
a
v
b
y
P
FIGURE 3
SOLUTION
(a)
This is a simple application oF the Law oF Cosines. Connect points
A
and
B
in the diagram to Form a line segment
that we will call
f
. Then, the Law oF Cosines says that
f
2
=
x
2
+
y
2
−
2
cos
. By taking square roots, we fnd that
f
=
p
x
2
+
y
2
−
2
cos
.
(b)
Using the chain rule,
df
dt
=
∂
f
∂
x
dx
+
∂
f
∂
y
dy
so we get
=
(
x
−
y
cos
)
/
p
x
2
+
y
2
−
2
cos
+
(
y
−
x
cos
)
/
p
x
2
+
y
2
−
2
cos
and using
x
=
30,
y
=
20, and
/
=
4,
/
=
3, we get
=
180
−
170 cos
√
1300
−
1200 cos
Further Insights and Challenges
36.
The Law oF Cosines states that
c
2
=
a
2
+
b
2
−
2
ab
cos
,where
a
,
b
,
c
are the sides oF a triangle and
is the angle
opposite the side oF length
c
.
(a)
Use implicit diFFerentiation to compute the derivatives
∂
∂
a
,
∂
∂
b
,and
∂
∂
c
.
(b)
Suppose that
a
=
10,
b
=
16,
c
=
22. Estimate the change in
iF
a
and
b
are increased by 1 and
c
is increased by 2.
(a)
Let
F
(
a
,
b
,
c
,
)
=
a
2
+
b
2
−
2
cos
−
c
2
. We use the Formulas obtained by implicit diFFerentiation (Eq. (6)) to
write
∂
∂
a
∂
F
∂
a
∂
F
∂
,
∂
∂
b
∂
F
∂
b
∂
F
∂
,
∂
∂
c
∂
F
∂
c
∂
F
∂
(1)
The partial derivatives oF
F
are
∂
F
∂
a
=
2
a
−
2
b
cos
,
∂
F
∂
b
=
2
b
−
2
a
cos
,
∂
F
∂
c
2
c
,
∂
F
∂
=
2
sin
Substituting these derivatives in (1), we obtain
∂
∂
a
2
a
−
2
b
cos
2
sin
a
−
b
cos
sin
∂
∂
b
2
b
−
2
a
cos
2
sin
b
−
a
cos
sin