# Untitled Page 9

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## 2.4 The chain rule

Figure i shows three clowns on seesaws. If the leftmost clown moves down by a distance *dx*,
the middle one will come up by *dy*, but this will also cause the one on the right to move down by *dz*.
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 *x* causes *y* to change, and *y* then causes *z* to change,
then this chain of changes has a cascading effect. Mathematically,
there is no big mystery here. We simply cancel *dy* 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 *dy*/*dx* 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 on page 151.

##### Example 9

◊ Jane hikes 3 kilometers in an hour, and hiking burns 70 calories per kilometer. At what rate does she burn calories?

◊ We let *x* be the number of hours she's spent hiking so far, *y* the distance covered,
and *z* the calories spent. Then

##### Example 10

◊ Figure j 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 11

◊ Find the derivative of the function *z*(*x*)=sin(*x*^{2}).

◊ Let *y*(*x*)=*x*^{2}, so that *z*(*x*)=sin(*y*(*x*)). 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 *y*.

##### Example 12

Let's express the chain rule without the use of the Leibniz notation. Let the function*f*be defined by

*f*(

*x*)=

*g*(

*h*(

*x*)). Then the derivative of

*f*is given by

*f*'(

*x*)=

*g*'(

*h*(

*x*))⋅

*h*'(

*x*).

##### Example 13

◊ We've already proved that the derivative of*t*

^{k}is

*kt*

^{k-1}for

*k*=-1 (example 10 on p. 27) and for

*k*=1, 2, 3, ... (p. 140). Use these facts to extend the rule to all integer values of

*k*.

◊
For *k*<0, the function *x*=*t*^{k} can be written as *x*=(*t*^{-1})^{-k}, where -*k* is positive.
Applying the chain rule, we find *dx*/*dt*=(-*k*)(*t*^{-1})^{-k-1}(-*t*^{-2})=*kt*^{k-1}.