Calculus: Analytic Geometry and Calculus, with Vectors

3.5 Difference quotients and derivatives 163

and hence

(5) f(x + .x) - f(x) = g(x + Ex)h(x + Ax) - g(x)h(x).

To make the right side more tractable, we subtract and add the term g(x + Ox)h(x)
and then divide by Ox to obtain

(6) f (x +

Ax)

• f (x) = g(x + Ax)

h(x + Ax) - h(x)

Ox Ax

+ h(x)g(x + X) - g(x)

But the hypotheses of our theorem, the definition of derivative, and Theorem

3.58 imply that

(7) lim g(x + Ax) = g(x), lim

h(x

+ z - h(x)= h'(x),

p-,Qlim g(x + Ax) - g(x)Ax - g'(x)

It follows that the limit, as, x approaches zero, of the right member of (6) is the

right member of the formula

(8) lliimof(x +AX)- f(x)

AX = g(x)h'(x) + h(x)g'(x)

The limit of the left member must be the same. Therefore, (8) holds, and (3)

then follows from the definition of the derivative of f at x. This proves the

theorem. This proof is essentially the same as the proof involving (3.57). If

we set u = g(x), v = h(x), y = f(x), u + Au = g(x + Ax), V + AV = h(x + dx),

and y + Ay = f(x + Ax), then (6) becomes

(9) Ax = (u + Au)Ox +vOx

which is, except for a minor shuffling of terms, the same as (3.57). The version

involving (3.57) is usually preferred in elementary courses because the formulas

involving Du, Ov, and Ay flow more smoothly and quickly than those given above.

19 Remark: As was said in passing in the text, discussions of names and sym-

bols can be long and perhaps dismal. We call a rose "a rose" because everyone

else does, and we do not need another reason. We call the number

(1) lim Ax + Ax)-AX)

Ago Ox

when it exists, "the derivative of f at x" because everyone else does, and we do

not need another reason. We can denote this number by f'(x) because everyone

else does, and we do not need another reason. If we want to know what f'(x)

means, we do not look at f' (x); we look at the definition of f'(x) and see that

(2) f'(x) = lim

f(x +Ax) - f(x)

Ax-4o Ax