Chain Rule

If we have a function z = f(y), where y is in turn a function of another variable x, say, y = g(x), then the derivative of z with respect to jc is equal to the derivative of z with respect to y, times the derivative of y with respect to jc. Expressed symbolically,

This rule, known as the chain rule, appeals easily to intuition. Given a Ax, there must result a corresponding Ay via the functiony = g(x), but this Ay will in turn bring about a Az via the function z =/(>')• Thus there is a "chain reaction" as follows:

The two links in this chain entail two difference quotients, Ay/Ax and Az/Ay, but when they are multiplied, the Ay will cancel itself out, and we end up with

'Az Ay Ay Ax a difference quotient that relates Az to Ax. If we take the limit of these difference quotients as Ax ->_0_ (which implies Ay -* 0), each difference quotient will turn into a derivative; i.e., we shall have {dz / dy)(dy / dx) = dz/dx. This is precisely the result in (7.11).

In view of the function y = g(x), we can express the function z = f(y) as z = f[g(x)], where the contiguous appearance of the two function symbols/ and g indicates that this is a composite function (function of a function). It is for this reason that the chain rule is also referred to as the composite-function rule or function-of-a-function rule.

The extension of the chain rule to three or more functions is straightforward. If we have z = f(y), y = g(x), and x = h(w), then dz dz dy dx , , ,, ,,,, , and similarly for cases in which more functions are involved. Example 1 If z = 3y2, where}' = 2x + 5, then f-l^-w-u.-ja^)

Example 3 The usefulness of this rule can best be appreciated when we must differentiate a function such as z = (x1 + 3x - 2)17. Without the chain rule at our disposal, dz/dx can be found only via the laborious route of first multiplying out the 17th-power expression. With the chain rule, however, we can take a shortcut by defining a new, intermediate variabley — x2 + 3x - 2, so that we get in effect two functions linked in a chain:

z = yxl and y = x2 + 3x - 2 The derivative dz/dx can then be found as follows:

Example 4 Given a total-revenue function of a firm R = f(Q), where output is a function of labor input L, or Q = g(L), find dR/dL. By the chain rule, we have _______

Translated into .economic terms^ dR/dQ is the MR func^gjp and dO/dL is the marginal-physical-produpt-of-labor (MPPl) functign. Similarly, dR/dL has the connotation of the marginal-revenue-product-of-labor (MRP^j function. Thus the result shown above constitutes the mathematical statement of the well-known result in economics that MRP, = MR • MPP, .

If the function y = /(x) represents a one-to-one mapping, i.e., if the function is such that a different value of x will always yield a different value of y, the function / will have an inverse function x = f~1 (y) (read: "x is an inverse function of y "). Here, the symbol f~1 is a function symbol which, like the derivative-function symbol /', signifies a function related to the function /; it does woijnean the reciprocal of the function/(x).

What the existence of an inverse function essentially means is that, in this case, not only will a given value of x yield a unique value of y [that is, y = /(x)], but also a given value of y will yield a unique value of x. To take a nonnumerical instance, we may exemplify the one-to-one mapping by the mapping from the set of all husbands to the set of all wives in a monogamous society. Each husband has a unique wife, and each wife has a unique husband. In contrast, the mapping from the set of all fathers to the set of all sons is not one-to-one, because a father may have more than one son, albeit each son has a unique father.

When x and y refer specifically to numbers, the property of one-to-one mapping is seen to be unique to the class of functions known as monotonic functions. Given a function /(x), if successively larger values of the independent variable x always lead to successively larger values of /(x), that is, if x,>x2 /(*,)>/(*2) then the function f is said to be an increasing (or monotonically increasing)

function.* If successive increases in x always lead to successive decreases in f(x), _that is, if ~

on the other hand, the function is said to be a decreasing (or monotonically decreasing) function. In either of these cases, an inverse function f~1 exists.

A practical way of ascertaining the monotonicity of a given functiony = f(x) is to check whether the derivative f'(x) always adheres to the same algebraic sign (not zero) for all values of x. Geometrically, this means that its slope is either always upward or always downward. Thus a firm's demand curve Q = f(P) that has a negative slope throughout is monotonic. As such, it has an inverse function P = f~l(Q), which, as mentioned previously, gives the average-revenue curve of the firm, since P = AR.

Example 5 The function

has the derivative dy/dx = 5, which is positive regardless of the value of x; thus the function is monotonic. (In this case it is increasing, because the derivative is positive.) It follows that an inverse function exists. In the present case, the inverse function is easily found by solving the given equation, / = 5x + 25 for x. The result is the function ^ _ -f _ ^ y.

It is interesting to note that this inverse function is also monotonic, and increasing, because dx/dy = y > 0 for all values of/.

Generally speaking, if an inverse function exists, the original and the inverse functions must both be monotonic. Moreover, if / 1 is the inverse function of /, then / must be the inverse function of f '; that is, f and /"' must be inverse functions of each other.

It is easy to verify that the graph of y = f(x) and that of x = f~ '(>>) are one_ and the same, only with the axes reversed. If one lays the x axis of the /"1 graph over the x axis of the / graph (and similarly for the y axis), the two curves will coincide. On the other hand, if the x axis of the /"1 graph is laid over the / axis of

* Some writers prefer to define an increasing function as a function with the property that xx>x2 =» f(xt)-f(x2) [with a weak inequality]

and then reserve the term strictly increasing function for the case where xt > x2 =» f(x\)>f(x2) [with a strict inequality]

Under this usage, an ascending step function qualifies as an increasing (though not strictly increasing) function, despite the fact that its graph contains horizontal segments. We shall not follow this usage in the present book. Instead, we shall consider an ascending step function to be, not an increasing function, but a nondecreasing one. By the same token, we shall regard a descending step function not as a decreasing function, but as a nonincreasing one.

the / graph (and vice versa), the two curves will become mirror images of each other with reference to the 45^ line drawn through the origin. This mirror-image relationship provides us with an easy way of graphing the inverse function f ', once the graph of the original function / is given. (You should try this with the two functions in Example 5.)

For inverse functions, the rule of differentiation is dx = 1 _ dy dy/dx ' c/y

This means that the derivative of the inverse function is the reciprocal of the derivative of the original function; as such, dx/dy must take the same sign as dy/dx, so that if /is increasing (decreasing), then so must be/1.

As a verification of this rule, we can refer back to Example 5, where dy/dx was found to be 5, and dx/dy equal to These two derivatives are indeed reciprocal to each other and have the same sign.

In that simple example, the inverse function is relatively easy to obtain, so that its derivative dx/dy can be found directly from the inverse function. As the next example shows, however, the inverse function is sometimes difficult to express explicitly, and thus direct differentiation may not be practicable. The usefulness of the inverse-function rule then becomes more fully apparent.

Example 6 Given y = x5 + x, find dx/dy. First of all, since f=5*4+>>° k for any value of x, the given function is monotonically increasing, and an inverse function exists. To solve the given equation for x may not be such an easy task, but the derivative of the inverse function ca^nevertheless (be found quickly by use of the inverse-function rule:

The inverse-function rule is, strictly speaking, applicable only when the function involved is a one-to-one mapping. In fact, however, we do have some leeway. For instance, when dealing with a U-shaped curve (not monotonic), we may consider the downward- and the upward-sloping segments of the curve as representing two separate functions, each with a restricted domain, and each being monotonic in the restricted domain. To each of these, the inverse-function rule can then again be applied.

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