

Figure 2.9 shows three
clowns on seesaws. If the leftmost clown moves down by a distance ,
the middle one will come up by
, but this will also cause the one on
the right to move down by
. If we want to predict how much the
rightmost clown will move in response to a certain amount of motion by the leftmost one, we have
This is called the chain rule. It says that if a change in causes
to change, and
then
causes
to change, then this chain of changes has a cascading effect.
Mathematically, there is no big mystery here. We simply cancel
on the
top and bottom. The only minor subtlety is that we would like to be able to be sloppy by using an expression like
to mean both the quotient of two infinitesimal numbers and a derivative, which is defined as the standard part of this quotient.
This sloppiness turns out to be all right, as proved in Proof
of the chain rule.
Example
Jane hikes 3 kilometers in an hour, and hiking burns 70 calories per kilometer. At what rate does she burn calories?
We let be the number of hours she’s spent hiking so far,
the distance covered, and
the calories spent. Then
Example
Figure 2.10 shows a piece of farm equipment containing a train of gears with 13, 21, and 42 teeth. If the smallest gear is
driven by a motor, relate the rate of rotation of the biggest gear to the rate of rotation of the motor.
Let x, y, and z be the angular positions of the three gears. Then
by the chain rule,

The chain rule lets us find the derivative of a function that has been built out of one function stuck inside another.
Example
Find the derivative of the function .
Let , so that
. Then
The way people usually say it is that the chain rule tells you to take the derivative of the
outside function, the sine in this case, and then multiply by the derivative of “the inside stuff,” which here is the square. Once you get used to doing it, you don’t need to invent a third,
intermediate variable, as we did here with
.
Example
Let’s express the chain rule without the use of the Leibniz notation. Let the function be defined by
. Then the derivative of
is given by
Example
We’ve already proved that the derivative of is
for
(Example) and for
(Derivatives of polynomials). Use these facts to extend the
rule to all integer values of
.
For , the function
can be written as
, where
is positive. Applying the chain rule, we find
.
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