252
III - Convergence: Continuous variables
lb f(x)dx = 10gb -loga.
Let us take b = a + h, with h > 0 for simplicity. Now f(x) remains between
I/(a + h) and lla between a and b; since the interval (a, a + h) is of length h
the integral thus lies between hl(a + h) and hla, whence
I/(a + h) :::; [log(a + h) -logaJlh :::; lla.
As h tends to 0, the outer terms converge to 1 I a. Similar calculation when h
tends to 0 through negative values. Thus we obtain the relation
(14.15)
log' x = llx
of Chap. II, nO 10, which we proved in another way in Chap. IV and which
we accepted in Chap. II, nO 5, for x = 1, to obtain or suggest the formula
log x = lim n (xl/n - 1) .
15 - Rules for calculating derivatives
If f'(a) exists for every a E I, we say that f is differentiable on I; the map
f' : x 1--+ f'(x) is the derived function of f. It may happen that the latter is
again differentiable, whence the concept of second derivative f" = (f')', and
so on. If one can continue indefinitely one says that f is indefinitely differentiable or of class coo on I. More generally, f is said to be p times continuously
differentiable or of class CP on I if the derived functions 1', f", ... ,f Cp ) exist
and are continuous on I. There are more subtle concepts, but these suffice in
the great majority of cases.
Derivatives obey simple rules that the reader surely knows already.
(D 1) If I and g are differentiable at a then so is their sum
sex) = I(x) + g(x), and s'(a) = 1'(a) + g'(a).
Indeed
(15.1)
I(a + h) = f(a) + I'(a)h + o(h),
g(a + h)
g(a) + g'(a)h + o(h),
with two functions o( h), not the same. On adding,
sea + h) = sea) + ch + o(h) + o(h)
where c = f'(a) + g'(a). It remains to show that