§4. Differentiable functions
249
One can understand this more easily in the following way. In the formula (3), the difference between the right hand side and c tends to 0 with h.
Multiplying by h, one has a relation
(14.12)
f(a + h) = f(a) + f'(a)h + o(h) when h -+ 0,
where the symbol o(h), with a lower case 0 like Newton, curiously - but here
it is the h multiplying f' (a) which plays the role of his 0 -, denotes (Chap. II,
nO 4) a function about which one can say no more than that
(14.13)
limo(h)jh = 0,
in other words such that, for any r > 0 there exists an r' > 0 for which
(14.14)
Ihl < r' ===> lo(h)1 < rlhl·
Conversely, the relation
f(a + h) = f(a) + ch + o(h) when h -+ 0,
where c is a constant, proves that f admits a derivative equal to c at a;
indeed, the preceding relation can be rewritten as
f(a + h) - f(a) _ c = o(h)jh
h
with a right hand side which, by definition, tends to 0, as we have already
explained in Chap. II, n° 4.
Like Newton's, this notation may serve to deduce a relation between the
derivatives from a relation between two functions x(t) and y(t). Take for
example the relation x 2 + y2 = 1. Replacing x(t) by x(t + h), i.e. by x(t) +
x'(t)h + o(h), and y(t) by y(t + h) = ... , one clearly finds [writing x, x', . ..
instead of x(t), x'(t), . .. J a relation
0= 2xx'h + 2yy'h + ...
where the ... denote terms containing o(h) or even o(h)2 as a factor. On
dividing throughout by h there follows a relation of the form
° = 2xx' + 2yy' + terms which tend to ° with h.
So 2xx' + 2yy' = 0, since these are the terms independent of h.
Newton's method is a little simpler since, following him, the two terms
x'(t)h + o(h) are condensed into just xo; equivalently, one eliminates the
residual term o(h) and replaces x by x+x'h, y by y+y'h, etc. in the equations.
249
One can understand this more easily in the following way. In the formula (3), the difference between the right hand side and c tends to 0 with h.
Multiplying by h, one has a relation
(14.12)
f(a + h) = f(a) + f'(a)h + o(h) when h -+ 0,
where the symbol o(h), with a lower case 0 like Newton, curiously - but here
it is the h multiplying f' (a) which plays the role of his 0 -, denotes (Chap. II,
nO 4) a function about which one can say no more than that
(14.13)
limo(h)jh = 0,
in other words such that, for any r > 0 there exists an r' > 0 for which
(14.14)
Ihl < r' ===> lo(h)1 < rlhl·
Conversely, the relation
f(a + h) = f(a) + ch + o(h) when h -+ 0,
where c is a constant, proves that f admits a derivative equal to c at a;
indeed, the preceding relation can be rewritten as
f(a + h) - f(a) _ c = o(h)jh
h
with a right hand side which, by definition, tends to 0, as we have already
explained in Chap. II, n° 4.
Like Newton's, this notation may serve to deduce a relation between the
derivatives from a relation between two functions x(t) and y(t). Take for
example the relation x 2 + y2 = 1. Replacing x(t) by x(t + h), i.e. by x(t) +
x'(t)h + o(h), and y(t) by y(t + h) = ... , one clearly finds [writing x, x', . ..
instead of x(t), x'(t), . .. J a relation
0= 2xx'h + 2yy'h + ...
where the ... denote terms containing o(h) or even o(h)2 as a factor. On
dividing throughout by h there follows a relation of the form
° = 2xx' + 2yy' + terms which tend to ° with h.
So 2xx' + 2yy' = 0, since these are the terms independent of h.
Newton's method is a little simpler since, following him, the two terms
x'(t)h + o(h) are condensed into just xo; equivalently, one eliminates the
residual term o(h) and replaces x by x+x'h, y by y+y'h, etc. in the equations.