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Section 14.4 The Chain Rule
Shubhodeep Mukherji
Chain Rule for a single variable, we said that when
w
=
f
(
x
)
was a diﬀerentiable function of
x
and
x
=
g
(
t
) was a diﬀerentiable
function of
t
,
w
became a diﬀerentiable function of
t
and
dw
dt
could
be calculated with the formula
dw
dt
=
dw
dx
dx
dt
.
For functions of two or more variables the Chain Rule has several
forms. The form depends on how many variables are involved but
works like Chain Rule for a single variable after accounting for pres
ence of additional variables.
1 Functions of Two Variables
If
w
=
f
(
x,y
) has continuous partial derivatives
f
x
and n
f
y
and
if
x
=
x
(
t
),
y
=
y
(
t
) are diﬀerentiable functions of
t
, then the
composite
w
=
f
(
x
(
t
)
,y
(
t
)) is a diﬀerentiable function of
t
and
df
dt
=
f
x
(
x
(
t
)
,y
(
t
))
*
x
0
(
t
) +
f
y
(
x
(
t
)
,y
(
t
))
*
y
0
(
t
)
,
or
dw
dt
=
∂f
∂x
dx
dt
+
∂f
∂y
dy
dt
Example 1
Applying the Chain Rule Find the derivative of
w
=
xy
with respect to
t
along the path
x
=
cos
(
t
)
,
y
=
sin
(
t
)
.
Solution
1
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View Full Documentdw
dt
=
∂w
∂x
dx
dt
+
∂w
∂y
dy
dt
=
∂
(
xy
)
∂x
d
dt
(
cos
(
t
)) +
∂
(
xy
)
∂y
d
dt
(
sin
(
t
))
=
y
(

sin
(
t
)) +
x
(
cos
(
t
))
= (
sin
(
t
))(

sin
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 Spring '11
 profeessor

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