Calculus: Analytic Geometry and Calculus, with Vectors

164 Functions, limits, derivatives

We can observe that the Newton notation in the formula (2) uses functional

notation in a thoroughly standard way; if f is a function and x is a number,

then the right side of (2), when it exists, determines the value of the function

f at x. If, for example, f'(x) = 2x for each real x, then f'(0) = 0, f'(6) = 12,

f'(x2) = 2x2, and f'(sin x) = 2 sin x. It could be presumed that one good symbol
for the number in (1) should be enough, but it is not enough. Even if there were
no other reason, we would still be required to know another symbol in order to
be able to read scientific literature and to converse with scientists. We must
know that we can set y = f(x), so that in a particular case we have y = x2, and
we can denote the derivative of f at x by the symbol

ddx

If we want to know

whatdydxmeans, we do not look atay;dxwe look at the definition ofy and finddx that,

in the particular case,

(3) dydx= 2x.

According to the definition,

dz

is the derivative of f at x, and the meaning of

dx is not changed when we read "dee y dee x" or "the derivative of y with respect

to x" or even "the derivative of y with respect to x at x." The assertion (3)

always means that the derivative of f at x is 2x, and weird ways of reading the

assertion do not change the meaning of the assertion. The meaning of the

assertion is not changed when we realize that a silly result is obtained by sup-

posing that the d's and the x and the y in (3) are numbers and canceling the d's

to get y/x = 2x. The meaning of the assertion is still unchanged when we

realize that we never put x = 6 in the two members of (3) to obtain

(4) dyd6 =12.

We do, however, allow ourselves the liberty of writing

(5)

z

dx =2x or dxx2 = 2x

to abbreviate the statement that if y = f(x), where f is the function for which

f(x) = x2, then the derivative of f at x is 2x. From a logical point of view, every-

thing we have done can be summarized very simply. If we want to know the

meaning of the word "quibble," we do not look at the word "quibble"; we look

at a definition. Let us then quit quibbling about the meaning of dx We can

conclude with a cheerful remark. Whenever we are likely to encounter diffi-

culties with the Leibniz notation, we can discard it and use the Newton notation.

3.6 The chain rule and differentiation of elementary functions

To be able to illustrate methods by which fundamental formulas for

derivatives are used, we suppose that we know the five fundamental