The Chain Rule
The chain rule is the rule for finding the derivative of composite functions. A composite
function is one formed in the following manner. We say
F
is the composite of
f
and
g
,
written
F
=
f
◦
g
, if
F
(
x
) =
f
(
g
(
x
)). So in the formula for
f
(
x
), we substitute for
x
the
formula for
g
(
x
).
Examples.
If
f
(
x
) =
x
2
+3
x
+4 and
g
(
x
) = 5
x
+6 then
f
(
g
(
x
)) = (5
x
+6)
2
+3(5
x
+6)+4 =
25
x
2
+ 75
x
+ 58
.
Compare
g
(
f
(
x
)) = 5(
x
2
+ 3
x
+ 4) + 6 = 5
x
2
+ 15
x
+ 6.
Let
f
(
x
) =
√
x
and
g
(
x
) = 25

x
2
.
Then
F
(
x
) =
f
(
g
(
x
)) =
p
25

x
2
and
g
(
f
(
x
)) =
25

(
√
x
)
2
= 25

x
(but note, the domain of
g
(
f
(
x
)) is
x
≥
0).
Let
F
(
x
) = 110 sin(120
πx
). This is the composite of
f
(
x
) = 110 sin
x
and
g
(
x
) = 120
πx
.
The function
F
(
x
) =
1 +
x
1

x
¶
5
is the composite of
f
(
x
) =
x
5
and
g
(
x
) =
1 +
x
1

x
.
Here is another way of writing compositions: If
y
=
f
(
u
) and
u
=
g
(
x
) then
y
=
f
(
g
(
x
)).
For example, we may express
F
(
x
) =
1 +
x
1

x
¶
5
by writing
y
=
u
5
and
u
=
1 +
x
1

x
.
The chain rule.
Suppose
F
(
x
) =
f
(
g
(
x
)). Then
F
0
(
x
) =
f
0
(
g
(
x
))
g
0
(
x
).
Examples.
Let
F
(
x
) = (
x
2
+ 1)
6
. Here,
f
(
x
) =
x
6
and
g
(
x
) =
x
2
+ 1, so
f
0
(
x
) = 6
x
5
and
g
0
(
x
) = 2
x
. Also,
f
0
(
g
(
x
)) = 6(
x
2
+ 1)
5
. So
F
0
(
x
) =
f
0
(
g
(
x
))
g
0
(
x
) = 6(
x
2
+ 1)
5
(2
x
).
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