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2.4. The Chain Rule

Figure I. Three clowns on seesaws demonstrate 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

\[\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx} .\]

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 17

◊ 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

\begin{align} \frac{dz}{dx} &= \left(\frac{70 \text{cal}}{1 \text{km}}\right) \left(\frac{3 \text{km}}{1 \text{hr}}\right) \\ &= 210 \text{cal}/\text{hr} . \end{align}

Example 18

Figure J. Example 18.

◊ 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,

\begin{align} \frac{dz}{dx} &= \frac{dz}{dy} \cdot \frac{dy}{dx} \\ &= \frac{13}{21} \cdot \frac{21}{42} \\ &= \frac{13}{42} . \end{align}

The chain rule lets us find the derivative of a function that has been built out of one function stuck inside another.

Example 19

◊ Find the derivative of the function z(x)=sin(x2).

◊ Let y(x) = x2, so that z(x) = sin(y(x)). Then

\begin{align} \frac{dz}{dx} &= \frac{dz}{dy} \cdot \frac{dy}{dx} \\ &= \cos(y) \cdot 2x \\ &= 2x \cos(x^2) \end{align}

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 20

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 21

◊ We've already proved that the derivative of tk is ktk-1 for k = -1 (Example 10) 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=tk 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) = ktk-1.


[Page number refers to the pdf version of this book.]