§4. Differentiable functions
259
it must not be applied too mechanically since the points where the derivatives
are to be calculated are not indicated. Formulae of this kind, not appearing
in SO convenient a form chez Newton, led to the success of Leibniz' system,
relieving the users from having to think, though not, even nowadays, from
sometimes writing stupidities.
There are also expressions analogous to df or dy j dx for the second differentials and derivatives ... , appreciably more subtle and, in fact, too subtle
and useless to have had lasting success in the elementary theory of functions
of one real variable, except as a convenient notation. Consider the linear function df(x) : h f-+ df(x; h) as being itself a function of x, namely df : x f-+ df(x);
although the values of this function are not numbers 34 , one can calculate its
differential d(df)(a) at a point a, and naturally one denotes it by d 2 f(a); to
find it one calculates
(15.9)
df(a + k) - df(a),
an expression which denotes the function
(15.10)
h f-+ df(a + k; h) - df(a; h) = J'(a + k)h - J'(a)h;
88 k tends to zero
(15.11)
j'(a + k) - j'(a) = j"(a)k + o(k),
so that (9) is the sum of the two following functions of h: (i) the function
h f-+ f"(a)kh, (ii) a function o(k)h = o(kh). Thus the differential d 2 f(a),
which involves the variable k for the first d sign and the variable h for the
second, is given by
(15.12)
d 2 f(a) : (h, k) 1----+ j"(a)hk = j"(a)dx(a; h)dx(a; k)
by (14.6) or, in short, d 2 f(a) = f"(a)dx(a)2 [up to the fact that the usual
square of a function of h is a function of h and not of the pair (h, k) ... J. Hence
Leibniz' expression d 2 yjdx 2 to denote the second derivative of y = f(x). Another explanation of the notation is that the second derivative is df' j dx, and
since f' = dyjdx one "clearly" finds d 2 yjdx 2 . On this point, it would be
considerably clearer once and for all to write D for the derivation operator 35
djdx, i.e. the map f f-+ 1', and D2 = DoD for the map f f-+ f", as already
said above.
34 For readers who know the elements of linear algebra, these are linear forms on
R considered as a real vector space of dimension 1. This set, endowed with the
obvious algebraic operations (addition, product by a scalar c E R) is in turn a
vector space of dimension 1. There is always a tendency to confuse it with R
because a linear form on R, i.e. a function h f-+ ch, is characterised by a number
c E R.. But a real number is not a function defined on R..
3& The word "operator", which is also used in algebra ("linear operator") and elsewhere in analysis ("Laplace operator" for example) is synonymous with the words
"function" and ''map'', although one frequently says "operators" without prescribing exactly the sets where they are defined or take their values.
259
it must not be applied too mechanically since the points where the derivatives
are to be calculated are not indicated. Formulae of this kind, not appearing
in SO convenient a form chez Newton, led to the success of Leibniz' system,
relieving the users from having to think, though not, even nowadays, from
sometimes writing stupidities.
There are also expressions analogous to df or dy j dx for the second differentials and derivatives ... , appreciably more subtle and, in fact, too subtle
and useless to have had lasting success in the elementary theory of functions
of one real variable, except as a convenient notation. Consider the linear function df(x) : h f-+ df(x; h) as being itself a function of x, namely df : x f-+ df(x);
although the values of this function are not numbers 34 , one can calculate its
differential d(df)(a) at a point a, and naturally one denotes it by d 2 f(a); to
find it one calculates
(15.9)
df(a + k) - df(a),
an expression which denotes the function
(15.10)
h f-+ df(a + k; h) - df(a; h) = J'(a + k)h - J'(a)h;
88 k tends to zero
(15.11)
j'(a + k) - j'(a) = j"(a)k + o(k),
so that (9) is the sum of the two following functions of h: (i) the function
h f-+ f"(a)kh, (ii) a function o(k)h = o(kh). Thus the differential d 2 f(a),
which involves the variable k for the first d sign and the variable h for the
second, is given by
(15.12)
d 2 f(a) : (h, k) 1----+ j"(a)hk = j"(a)dx(a; h)dx(a; k)
by (14.6) or, in short, d 2 f(a) = f"(a)dx(a)2 [up to the fact that the usual
square of a function of h is a function of h and not of the pair (h, k) ... J. Hence
Leibniz' expression d 2 yjdx 2 to denote the second derivative of y = f(x). Another explanation of the notation is that the second derivative is df' j dx, and
since f' = dyjdx one "clearly" finds d 2 yjdx 2 . On this point, it would be
considerably clearer once and for all to write D for the derivation operator 35
djdx, i.e. the map f f-+ 1', and D2 = DoD for the map f f-+ f", as already
said above.
34 For readers who know the elements of linear algebra, these are linear forms on
R considered as a real vector space of dimension 1. This set, endowed with the
obvious algebraic operations (addition, product by a scalar c E R) is in turn a
vector space of dimension 1. There is always a tendency to confuse it with R
because a linear form on R, i.e. a function h f-+ ch, is characterised by a number
c E R.. But a real number is not a function defined on R..
3& The word "operator", which is also used in algebra ("linear operator") and elsewhere in analysis ("Laplace operator" for example) is synonymous with the words
"function" and ''map'', although one frequently says "operators" without prescribing exactly the sets where they are defined or take their values.