Sum Difference Rule

The derivative of a sum (difference) of two functions is the sum (difference) of the derivatives of the two functions:

~\f(x)±g(x)]= ^-f(x)±^j-x(x) = f{x)±g/(x) dx dx dx

The proof of this again involves the application of the definition of a derivative and of the various limit theorems. We shall omit the proof and. instead, merely verify its validity and illustrate its application.

From the function y= 14x\ we can obtain the derivative dyjdx — Alx1. But 14x3 = Sx* -9x\ so that y may be regarded as the sum of two functions f(x)-5x3 and g(x) = 9x3. According to the sum rule, we then have

dx dx ax dx which is identical with our earlier result.

This rule, which we stated in terms of two functions, can easily be extended to more functions- Thus, it is also valid to write

The function cited in Example 1, y = 14x3, can be written as y = 2x3 + 1 3x3 - x3. The derivative of the latter, according to the sum-difference rule, Is

dx dx which again checks with the previous answer.

This rule is of great practical importance. With it at our disposal, it is now possible to find the derivative of any polynomial function, since the latter is nothing but a sum of power functions.

~(7xA + 2xl - 3* + 37) = 28x3 + 6x2 - 3 + 0 = 28x3 + 6xz - 3 dx

Note that in Examples 3 and 4 the constants c and 37 do not really produce any effect on the derivative, because the derivative of a constant term is zero. In contrast to the multiplicative constant, which is retained during differentiation, the additive constant drops out, This fact provides the mathematical explanation of the well-known economic principle that the fixed cost of a firm does not affect its marginal cost. Given a shorl-run total-cost function the marginal-cost function (for infinitesimal output change) is the limit of the quotient AC/AQ, or the derivative of the C function:

whereas the fixed cost is represented by the additive constant 75. Since the latter drops out during the process of deriving dC/dQ, the magnitude of the fixed cost obviously cannot affect the marginal cost.

In general, if a primitive function y — fix) represents a total function, then the derivative function dv/dx is its marginal function. Both functions can, of course, be plotted against the variable :t graphically; and because of the correspondence between the derivative of a function and the slope of its curve, for each value ofx the marginal function should show the slope of the total function at that value ofx. In Fig. 7.1a, a linear (constant-slope) total function is seen to have a constant marginal function. On the other hand, the nonlinear (varying-slope) total function in Fig. 7.16 gives rise to a curved marginal function, which lies below (above) the horizontal axis when the total function is negatively (positively) sloped. And, finally, the reader may note from Fig. 7.1c (cf. Fig. 6,5) that



"nonsmoothncss" of a total function will result in a gap (discontinuity) in the marginal or derivative function. This is in sharp contrast to the everywhere-smooth total function in Fig. 7.1/> which gives rise to a continuous marginal function. For this reason, the smoothness of a primitive function can be linked to the continuity of its derivative function. In particular, instead of saying that a ccrtain function is smooth (and differentiate) everywhere, we may alternatively characterize it as a function with a continuous derivative function, and refer to it as a continuously differentiahle function.

The following notations are often used to denote the continuity and the continuous differentiability of a function /:

/ € C[V] or / e C'\ f is continuously differentiate where Cco\ or simply C, is the symbol for the set of all continuous functions, and C[] \ or C\ is the symbol for the set of all continuously differcntiable functions.

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