Passing now to D's, we first attempt y = ex. Hence y + x=es TAx-=e- 11+ dx+ +...}.
1 2 Hence dy (eAx-1) z dx dx ex[ i x d x 2 13 This [ ] is term by term, except the first, less than sax, whose limit is 1; hence Lim. [ is 1: hence Lim. dy =ex, or Dex= ex. The ex x ponential ex is unchanged by Derivation as to its exponent. Hence Del = ea.'s:. Hence, if y= log x, ez = = 1, yx =e- or az D log x 1 . Hence D log u=. Hence, if y = log y m log x; Dy== For y = sin x, 2 d yin (x+2dx)sin x =2 cos (x- dx) sin dx; = cos (x dx) "; hencedx d sin ,cos x=sin x : hence D sin tr---os wit:. Hence D cos x= sin x 7 = cos ( X+ T) Hence derivation of sine and cosine as to the angle merely adds the angle. Also, D tan x=1-1- tan sec ace. If sin )=x, cos y = 1, yz-= 1 hence D . Similarly for If similarly D S 1a D sin D a a Similarly we treat the hyperbolic sine and cosine and tangent (lux, hcx, htx), and their inverses ht-'x, with the important results: Dhsxhcx, Max.-- 1 hte, 1 1 v + VT* 1 1 with easy generalization for u and formula especially important in Integration. By Deri vation the anti-transcendentals are thus reduced to algebraic forms, while the exponentials and goniometrics return into themselves; hence the inverse of Derivation, whatever it may be, applied to algebraic forms, may give rise to transcendentals. So much for ordinary alge braic and simply periodic functions.
The Infinite Series cannot always be differ entiated by differentiating term by term, but only under certain conditions of equable convergence. If each term fn(x) of
ft:) be unique and continuous and if I converge, for every x in [a, a + d], and if Lf(x)f(a), z.--a+0 for such a series the Theorem holds: If each term has a finite progressive DC, f.' (a), and if
dx)
the series of DCs, . eon n) verges equably for every Ix> 0 and
11:) In general, this narrower Theorem will an swer: If, for every x in [a, a+d], each f n(x) is unique, continuous, and has a D (and for za at least a progressive DC), and if both = and f'n (x) converge, the latter =o n o equably, then Lf(x)---f(a), and f(x) has a progressive DC at a, which is E fn'(a) ; i.e., we form the D of the infinite series by summing (to m ) the D's of the terms (for details see Real Variable, Theory of Functions of the.
For D of f(z) as to z, where sx+iy, see Complex Variable, Theory of Functions of.* A D of a first D is called a second D, written variously f"(y), Dxs, Di and so on for the 3d, 4th, ... nth D's. We see at once, Im sin sin Dn x= cos cos 2 + n 1 Dn ( 1)n (X m I M (X + A rational fraction must first be decomposed into such fractions. The exponential ex
repeats itself steadily, Dneax = aneax; hence 4)(D)eax = ib(a)eax, ito being algebraic and ra tional. The log x is at once reduced to a fraction by D log For a product uv we have Leibnites Theorem: + 2 1)-I + nun-wi -t- un--sh - precisely as in the Bionomial Theorem, the subscripts denoting the D's. For a quotient y= we write u=vy, +2n +trys, , un=vny nvn--1 . . . From these (n-1-1) simultaneous equations we form the eliminant of the n unknowns, y, i; this eliminant is linear in and yields (_onin u v 0 0 0.
YR tin +I fit v, V 0 .
th v, V1 V 0 . .
!ts Applications. 1. Let P be any ordinary point of a curve, S the foot of the ordinary y, T and N the feet (on X) of tangent and normal at P. Since yx = tan r, SN = y'yx, ST = y.xp, and we easily express PN, PT, etc. Also, if ito = angle of intersection of y=f(x) and Y=F(x), then tan If the curves touch, the 1 + yx numerator=0; if they are perpendicular, the denominator = 0.
2. Envelopes.Let F(x, y; be a system of curves distinguished by varying values of the parameter p. For any special value of p, F(x, y; p)'O will be one curve and F(x, y; p+ dp)::1 (2) a neighboring curve. Where do they meet? What relation connects x and y of the inter section, I, of any such pair? We must com bine (1) and (2) and eliminate p. If in the result we pass to the limit for dp 0, we shall find the locus of the intersection of consecutives (or the Envelope) of the system. For (2) we may put (1)-(2), or still better F (x, y; p dp) F (x, y; p) O. (3) dp It will be equivalent to invert the procedure, to pass to the limit and then eliminate p. So we get Fp(x, y; p) = 0, (4) between which and (1) we now eliminate p. This eliminant connects x and y for every intersection of two consecutives of the system, every instantaneous pivot about which the curve starts to turn into a neighboring position. But this is not all. It connects the x and y of all other points where meet two curves (branches) corresponding to the same p, as may thus be seen. Assign any pair of values to x and y, i. e., take any point in the plane, and ask what members of P=0 pass through it, i.e., what are the corresponding values of p? There are n such, if F be of nth degree in p. When will two of these p-roots be equal? Only when the p-discriminant of F=0 van ishes; i.e., when the eliminant of p between FCo and Fp=1) vanishes, as we know from Algebra. Hence this eliminant connects x and y for all points where meet two curves corresponding to the same or equal p's, This will include all cusps and nodes as well as instantaneous pivots; hence the p-eliminant=0 will be the equation of all cusp-loci and node loci as well as of envelope proper.