14.5: THE CHAIN RULE
KIAM HEONG KWA
We shall assume all functions are differentiable in this section.
1.
The Chain Rule
Suppose that
f
is a function of the
n
variables
x
1
,
x
2
,
· · ·
,
x
n
and
each
x
i
is a function of the
m
variables
t
1
,
t
2
,
c
· · ·
,
t
m
, then
(1.1)
∂f
∂t
j
=
n
X
i
=1
∂f
∂x
i
∂x
i
∂t
j
for each
j
,
j
= 1
,
2
,
· · ·
, m
. It should be noted that
•
∂f/∂t
j
is evaluated at (
t
1
, t
2
,
· · ·
, t
m
),
•
∂f/∂x
i
is evaluated at the corresponding value of (
x
1
, x
2
,
· · ·
, x
n
),
i.e. at
(
x
1
(
t
1
, t
2
,
· · ·
, t
m
)
, x
2
(
t
1
, t
2
,
· · ·
, t
m
)
,
· · ·
, x
n
(
t
1
, t
2
,
· · ·
, t
m
))
,
and
•
∂x
i
/∂t
j
is evaluated at (
t
1
, t
2
,
· · ·
, t
m
).
For instance, if
f
is a function of
x
,
y
, and
z
, while
x
,
y
, and
z
are
functions of
t
, then
df
dt
=
∂f
∂x
dx
dt
+
∂f
∂y
dy
dt
+
∂f
∂z
dz
dt
.
The functional dependence of these variables can be denoted by the
tree diagram
f
x
t
y
t
z
t
As another instance, if
f
is a function of
x
,
y
, and
z
, while
x
,
y
, and
z
are functions of
r
,
s
, and
t