Lagrange ((Theorie des I) at tains the notion of Derived Functions (or D's) by substituting x for x in F(x) = ao-F . . a nx-n , whence F (x ) m.=F(x) F' (x) F (n)(X) F(x) + • + • • • + In • en, where each F turns out to be formed from the preceding in the same way; they are the Derived Functions of F. A near-lying Gen 11.10 eralization considers f(x) an(x—a)s, n=o supposed absolutely convergent for all values of Jx—aj < -FR, i.e., for x within [a—R, a + R]. Then in the same [ ] all the co series — a) u•-• tn(m = 1, 2, 3, . . . ) n=xs in —In will also converge absolutely. Denote them in turn by r (x), f'(x),.. . Choose e so that fo) (x) = e jafl< R; then the series In =0 Im will also converge absolutely in the same [ and will equal f(x ). These Sums f(x), r (x), . . . are called 1st and 2d, . .. Derived Functions of f(x), which may be called its own 0th Derived Function. If instead of the inconvenient Lagrangian accents we put Cauchy's D's with proper exponents, we per ceive that these latter, denoting order of dif ferentiation, obey the same laws as ordinary exponents: Dm+n= Dom. Dn = Dn. Dm, etc.
It is usual, though not quite satisfactory, to denote the value of any derived function at any point (x = a) by writing a for x, thus: fix) (a). At this stage the D-notation is not so convenient. These special values are seen to be On finding hence the a's and substituting in the definition of Rs) .) we get f(x)= f(a)-f- * cof(n L (x a)ft x :-.0of(n) x ay% 11.0 In Such is the ordinary Taylor's Series, or laurin's (more justly Stirling's) in case a = O.
Lagrange supposed (amazingly, Picard) any arbitrary f(x) expansible in positive integral powers of (1--a), except for special values of a. However, presupposed only uniqueness and continuity in a definite interval, there may be no value of a in the interval for which such expansion is possible. Thus, for z -G 0 and =Al for x>0 cannot be developed in positive integral powers of x for x positive and p not integral. Hence this Lagrangian notion of derived function, while in general agreeing with the notion of D as limit of dif ference-quotient, is not yet so universal. • The notion of Differential, though unneces sary at this stage, is commonly introduced x) ( thus: From jfiz) = (x) As+ ads. This first part of df(x), namely, f'(x) dx, proportional to 41- and of the same order of infinitesimality, may be defined as the Differential of f(x) and may be denoted by df (x), which is thus a finite variable for zlxSO. For f x, we have dx dx, which is there dy _df (x ) fore diff erential of x. Hence —= (x),
dx i.e., the D of Az) as to x— the quotient of the dy di erentials of f (x) and x (Leibnitz). Here is strictly a fraction whose terms are by no means ((ghosts of departed quantities° ley). Geometrically, dy is the dy prolonged up to the tangent at P. change of the ordinate of the tangent when abscissa changes by dx; Lim. 4.1y —=1. This notion of differen dy tial, though useful in geometry, mechanics and elsewhere, rather embarrasses the theoietical development of the subject. Hence the terms Differentiation (=Derivation), to differentiate, and hence the names Differential Calculus, Differential Coefficient.
On these bases the structure of the DIFFER ENTIAL CALCULUS may now be safely erected. Primary formulae, easily established, are as follows (D meaning always Derivative as to x, u, v, etc., being simultaneous functions of x): D(* Du + Dv— Dw; D(uv)= Du -a • Du Dv= uv Dv)l (14 ± = te; C • ie.
Very important is Mediate Derivation, when y is function of a function of x, as y----4(u), u=f (x), hence y-i• j f (x) I (x). If then st and f have definite D's, itv(u) and f'(x), we have dy du hence -ux=010' (u) • f ( 4x 4u 4z' But yx may exist even when the fails, and this rule with it.
In particular, if y =f(x) and inversely x=ii(y) and if either variable has a D +0 dy so has the other. For if f (x) a Lim. —+ 0, d x ,dx dy dx 1 then hence -- dx — 1 x)" — i.e. in general, the D's of x as to y and of y as to x are reciprocals of each other: Xy=1.
If y'E1(x) be +, then x andy increase (or decrease) together, f(x) is called increasing at x. But if y be—, then x and y change oppo sitely, f (x) is decreasing at x.
Hence if f'(c) +0, f(x) must be > f(c) on one side of the point c, and< f(c) on the other. Hence Rolle's Theorem: If f(x) vanishes at a and b and has a D at every point within [a, b), then this D, f'(x) vanishes at some point within [4, b].
Now, f Wm' (b— a) (x) .1• (a) I — (x--a) .(b)-10 (a) 1 is such an f (x), made to order, ft. being differentiable within [a, b]. Hence f (x):÷:(b—a).1.' (x)— (b)-0 (a) must = 0 for some z in [a, b]. Hence 40(b)-10(a)=(b—a)C(i). Commonly we write a for x and x + h for b; then '1=x + Oh, where 8 is in [0, 1], so that (x+h)--0 (x)=1vp' (x-F0h), the extremely im portant formula for finite increments. Hence we see at once that if the D is everywhere 0 within an interval, the function is constant in that interval; and hence that two functions whose D's are equal in an interval can them selves differ only by a constant in that interval — a Theorem at the base of the Integral Calculus.