Calculus: Analytic Geometry and Calculus, with Vectors

166 Functions, limits, derivatives

Theorem 3.65 (chain rule) If f and g are functions such that g is

differentiable at x and f is differentiable at g(x) and if we set y = f(u) and
u = g(x) so that y = f(g(x)), then the chain formula

(3.66) dy=dy du=

dx dudxf'(u)g'(x) = f'(g(x))g'(x)

is valid at x.

To prove this theorem, we use the notation of the theorem to obtain

(3.661) Au = g(x + Ax) - g(x), Ay = f(u + Au) - f(u)

and observe that u and y are determined by x alone, while Au and Ay are

determined by x and Ax. Consider first the usual case in which there is a

number S1 such that 3, > 0 and Au 34 0 whenever 0 < JAxI < Si. Then,

when 0 < JAxI < Sl, we can write

(3.662) DY= DY Au =f(u + Au) - f(u) g(x + Ax) - g(x)

Ax Au Ax Au Ax

and, after observing that

(3.663) lim Au = lim - Ax =L0 = 0,

AZ-0 5,,o Ax dx

take limits as Ax approaches zero to obtain the required result. Because

division by zero is taboo, exceptional cases are more troublesome. We

can avoid this difficulty and handle all cases at once by setting

(3.664) c&(Au) =oy=f (U + Au) - f (u)

Du Du

(3.665) O(Au) = du = f'(u)

Then, whether Au is zero or not, we can write

(3.666) AY= cb(Du)

Du

= (Au)g(x + Ax) - g(x)

Ox Ox Ax

(Au 0 0)

(Au = 0).

and take limits as Ax approaches zero to obtain the required result.

The basic elementary functions can be separated into three classes.

The first class contains powers and roots of x, that is, functions of the

form x°, where a is a constant. The second class contains the six trigo-

nometric functions and the six inverse trigonometric functions. The

third class contains exponential functions of the form bx and logarithmic

functions of the form logb x, the base b being a constani. Thus there are

just 15 types of basic elementary functions. The class of elementary

functions includes the frightful function 4 having values

(3667) h(x) =log (1 + x2) + [ex + (x4 - 7x2 + sin-' 3x2)4].

sin e2' + e°'' as + x sine 4x - cos xb