1549901369-Elements_of_Real_Analysis__Denlinger_

306 Chapter 6 • Differentiable Functions

NOTATION FOR DERIVATIVES

It is somewhat cumbersome to continue using the subscript in x 0. We shall
usually write f'(x) instead of f'(x 0 ) whenever it is possible to do so without
ambiguity. Then the "derivative" becomes a function, f'. It is important to
understand that, in Definition 6.1.1, the derivative of a function f at a point
x 0 is a number, whereas we are now suggesting that we may also consider the
function f'. Of course, the two functions, f and f' may have different domains;
f(x) may exist where f'(x) does not.
Some of the common notational devices for derivatives are:

f'(x) = D x (f(x)) = d~ (f(x)) = d:~).

If y = f(x), then we can write

dy d

f'(x) = y' = - = -y.

dx dx

You are probably aware that calculus as a formal subject was developed
in the seventeenth and eighteenth centuries. One of the two principal inventors
of the subject, Sir Isaac Newton (English, 1642 - 1727) used a notation similar
to our y' for the derivative. The other principal inventor, Gottfried Wilhelm
Leibniz (German, 1646 - 1717) developed the dy/dx notation. Both notations
have advantages. The simplicity of y' is fr equently offset by the suggestive
power of the "differentials" in dy/dx.

ALGEBRA OF DERIVATIVES

Theorem 6.2.2 (Algebra of Derivatives) Suppose f and g are differen-
tiable at x and c E R Then

(a) cf is differentiable at x, and (cf)'(x) = c [f'(x)];

(b) f + g is differentiable at x, and (f + g)'(x) = f'(x) + g'(x);

(c) f - g is differentiable at x, and (f - g)'(x) = f'(x) - g'(x);

(d) Jg is differentiable at x, and (fg)'(x) = f(x)g'(x) + g(x)f'(x);

(e) ifg(x) =I-0, then[_ is differentiable atx, and (L)' (x) = g(x)f'(x) - f~x)g'(x).

g g (g(x))

[Rules (a)- (e) are called the constant multiple rule, sum rule, difference
rule, product rule, and quotient rule, respectively.]