on the right are best done with the Chain Rule.
Basically, the Chain Rule states that if
changes
times as fast as
and
changes
times as fast as
then
changes
times as fast as
EXAMPLE 1
The Derivative of a Composite Function
A set of gears is constructed, as shown in Figure 2.24, such that the second and third
gears are on the same axle. As the first axle revolves, it drives the second axle, which in
turn drives the third axle. Let
and
represent the numbers of revolutions per minute
of the first, second, and third axles. Find
and
and show that
Solution
Because the circumference of the second gear is three times that of the first,
the first axle must make three revolutions to turn the second axle once. Similarly, the
second axle must make two revolutions to turn the third axle once, and you can write
and
Combining these two results, you know that the first axle must make six revolutions
to turn the third axle once. So, you can write
.
In other words, the rate of change of
with respect to
is the product of the rate of
change of
with respect to
and the rate of change of
with respect to
x
.
u
u
y
x
y
Rate of change of first axle
with respect to third axle
dy
du
du
dx
3
2
6
Rate of change of second axle
with respect to third axle
Rate of change of first axle
with respect to second axle
dy
dx
du
dx
2.
dy
du
3
dy
dx
dy
du
du
dx
.
dy dx
,
du dx
,
dy du
,
x
y
,
u
,
x
.
dy du
du dx
y
x
,
du dx
u
u
,
dy du
y
y
x
tan
x
2
y
x
tan
x
y
3
x
2
5
y
3
x
2
y
sin 6
x
y
sin
x
y
x
2
1
y
x
2
1
With
the
Chain
Rule
Without
the
Chain
Rule
1
1
2
Axle 1
Axle 2
Axle 3
Gear 1
Gear 2
Gear 3
Gear 4
3
Axle 1:
revolutions per minute
Axle 2:
revolutions per minute
Axle 3:
revolutions per minute
Figure 2.24
x
u
y
Video
Animation
Try It
Exploration A

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SECTION 2.4
The Chain Rule
131
Example 1 illustrates a simple case of the Chain Rule. The general rule is stated
below.
Proof
Let
Then, using the alternative form of the derivative, you
need to show that, for
An important consideration in this proof is the behavior of
as
approaches
A problem occurs if there are values of
other than
such that
Appendix A shows how to use the differentiability of
and
to overcome this
problem. For now, assume that
for values of
other than
In the proofs
of the Product Rule and the Quotient Rule, the same quantity was added and sub-
tracted to obtain the desired form. This proof uses a similar technique—multiplying
and dividing by the same (nonzero) quantity. Note that because
is differentiable, it
is also continuous, and it follows that
as
When applying the Chain Rule, it is helpful to think of the composite function
as having two parts—an inner part and an outer part.
Outer function
Inner function
The derivative of
is the derivative of the outer function (at the inner function
)
times
the derivative of the inner function.
y
f
u
u
u
y
f u
y
f
g x
f
u
f
g
f
g c
g
c
lim
x
→
c
f g x
f g c
g x
g c
lim
x
→
c
g x
g c
x
c
lim
x
→
c
f g x
f g c
g x
g c
g x
g c
x
c
,
g x
g c
h
c
lim
x
→
c
f g x
f g c
x
c
x
→
c
.
g x
→
g c
g
c
.
x
g x
g c
g
f
g x
g c
.
c
,
x
,
c
.
x
g
h
c
f
g c
g
c
.