SECTION
15.6
The Chain Rule
(ET Section 14.6)
703
SOLUTION
The composite function
f
(
u
,v)
is obtained by replacing
u
and
v
in the formula for
f
(
u
by the corre-
sponding functions
u
=
rs
and
v
=
r
+
s
.Thisgives
f
(
u
(
r
,
s
), v(
r
,
s
)
)
=
u
(
r
,
s
)
e
v(
r
,
s
)
=
rse
r
+
s
Answer (a) is the correct answer.
3.
What is the value of
f
(
u
at
(
r
,
s
)
=
(
1
,
1
)
?
We compute
u
=
and
v
=
r
+
s
at the point
(
r
,
s
)
=
(
1
,
1
)
:
u
(
1
,
1
)
=
1
·
1
=
1
;
1
,
1
)
=
1
+
1
=
2
Substituting in
f
(
u
=
ue
v
,weget
f
(
u
¯
¯
¯
¯
(
r
,
s
)
=
(
1
,
1
)
=
1
·
e
2
=
e
2
.
4.
According to the Chain Rule,
∂
f
∂
r
is equal to (choose correct answer):
(a)
∂
f
∂
x
∂
x
∂
r
+
∂
f
∂
x
∂
x
∂
s
(b)
∂
f
∂
x
∂
x
∂
r
+
∂
f
∂
y
∂
y
∂
r
(c)
∂
f
∂
r
∂
r
∂
x
+
∂
f
∂
s
∂
s
∂
x
For a function
f
(
x
,
y
)
where
x
=
x
(
r
,
s
)
and
y
=
y
(
r
,
s
)
, the Chain Rule states that the partial derivative
∂
f
∂
r
is as given in (b). That is,
∂
f
∂
x
∂
x
∂
r
+
∂
f
∂
y
∂
y
∂
r
5.
Suppose that
x
,
y
,
z
are functions of the independent variables
u
,v,w
. Given a function
f
(
x
,
y
,
z
)
, which of the
following terms appear in the Chain Rule expression for
∂
f
∂w
?
(a)
∂
f
∂v
∂
x
(b)
∂
f
∂
x
(c)
∂
f
∂
z
∂
z
(d)
∂
f
By the Chain Rule, the derivative
∂
f
is
∂
f
=
∂
f
∂
x
∂
x
+
∂
f
∂
y
∂
y
+
∂
f
∂
z
∂
z
Therefore (c) is the only correct answer.