Constant Function Rule

The derivative of a constant function v = i, or j\x) — k, is identically zero, ix., is zero for all values of a:. Symbolically, this rule may be stated as: Given y = f(x) = k, the derivative is

Alternatively, we may state the rule as; Given y = f{x) = k, the derivative is where the derivative symbol has been separated into two parts, d/dx on the one hand, and y [or/(jr) or A] on the other. The first part, d/dx, is an operator symbol, which instructs us to perform a particular mathematical operation. Just as the operator symbol J instructs us to take a square root, the symbol d/dx represents an instruction to take the derivative of, or to differentiate, (some function) with respect to the variable jc. The function to be operated on (to be differentiated) is indicated in the second part; here it is v = f(x) — k.

The proof of the rule is as follows. Given fix) = k, we have f{N) = k for any value of N. Thus the value of fr(N)—the value of the derivative at* = Ar—as defined in (6,13) is fix) - f{N) k-k / (AO = Urn ^——- = lim -= lim 0 = 0

Moreover, since /Vrepresents any value of x at all, the result /'(N) = 0 can be immediately generalized lo f(x) = 0, This proves the rule.

It is important to distinguish clearly between the statement f'(x) = 0 and the similar-looking but different statement /'(jco) = By /'(*) = we mean that the derivative function f has a zero value for all values ofx: in writing f{xo) = 0, on the other hand, we are merely associating the zero value of the derivative with a particular value of*» namely, X = Xt).

As discussed before, the derivative of a function has its geometric counterpart in the slope of the curve. The graph of a constant function, say. a fixed-cost function C[? = f(Q) = SI„200. is a horizontal straight line with a zero slope throughout, Correspondingly, the derivative must also be zero for all values of Q\

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