dx Geometric Interpretation.—The Differential Calculus originated in the Problem of Tangents. Let P be any point of a curve referred to rectangular axes X, Y, and let P be between Q and Q'. Draw secants PQ, PQ', sloped 0 and a' to X, and to each other; draw ordinates through P, Q, Q'; through P and Q' draw parallels to X, meeting ordinates through Q and P at D and D'. Then PD=dx, D'Q'=d'x, DQ=dy, y Also — J =tan .4y = tan 6', tan (6' — 6) x tan O. If now by approaching Q and to P we can make and keep 0, and therefore tan 9, small at will, then the secants settle down into a common position called tangent the curve at P (sloped r to X) and the Jy 4 common limit of — and is tan r; or Jr Aix f' (x)=tan r. But if P were an angular point, then PQ and PQ' would not tend together, Jy would tend to one limit, the progressive dx 'y differential coefficient, and d — to another dx limit, the regressive differential coefficient; only when these two coalesce is there a five proper. Thus in y = (eT— 1) (e + i) , at the Origin, Lim. — Jy =-- e— +1) 4z the progressive DC-1, the regressive DC--1, the two limiting positions of the secants are perpendicular.
How thickly may such salients be strewn along a curve? To have a D. i.e., to be dif ferentiable. plainly the function must be continuous; it was long thought that this necessary condition was sufficient, that the continuous function possessed in general a D, save at certain special points. It was Riemann who first suggested (at least as early as 1861) the astonishing possibility that such an f(x) as sin though everywhere con 1 nl tinuous, was nowhere differentiable; but as he left no proof, it was generally thought he meant that it was possible to find such salients in every infinitesimal [x, x+ Jx], which was easy to show; but Weierstrass thought he meant strictly that the D did not exist for any value of x. In any case Weierstrass himself produced (18 July 1872) an example of such a function, y= I tog cos(aarx),— where a is an n-;) odd integer, b a number < 1, and ab > 1+ fir,— which, though everywhere con tinuous, has nowhere a D, since the progressive and regressive Difference-Quotients are every where opposite in sign and increase oppositely toward so as they pass over into Differential Coefficients (Math. Werke von K. Weierstrass, II,' p. 71-74). Geometrically, in the graph of the differentiable function, the polygon formed by n consecutive chords tends toward the curve for n=1-.co, PQ and PQ' tend to coalesce as Q and Q' both approach P, the triangle PQQ' becomes flatter and flatter (we may suppose the arc QPQ' steadily enlarged under a microscope to its original length as Q and Q' close down on P), the curve we may say is elementally straight at P. But with Wcierstrass's function
the polygon remains always re-entrant, a zag, and consecutive chords, PQ and PQ tend to separate at a straight angle. Such discontinuities may yet present themselves to the future student of nature.
= dy = 2x-lx Jx JY dy whence Dry s:: —yz=2.r. Here y is the area dx of a square whose side is x, and 2x is the border of the square perpendicular to which the square expands, the D is the front of variation. Similarly, if y =it?, =-- 27rx, the circum ference, the front of variation perpendicular to which x varies. If y =x", the front of variation perpendicular to which x varies. If y =1110, Dry = @x'= the sphere surface, the front of variation perpendicular to x. For y= x', D7y = 4x' = again the whole front of variation, though here our powers of envisagement fail us. Thus we are conducted to the Derivation of Assemblages, for which the reader must be referred to this latter subject.
Kinematic Let s= length of path of a moving point P, described in time t; Z Is and J t = corresponding changes in s and t; then = average speed during dt; ds Lim. =D es = instantaneous speed at speed at the instant t (i.e., end of t and ning of dt). There is no motion at the instant of time, nor at the point of path, but only during the time and space immediately about the point and instant. Instantaneous speed is a technical term for limit of average speed in the immediate vicinity of the point and instant. This instantaneous speed generally varies with t, and its D as to t is named accelera tion and is written product of this acceleration by the mass of the moving P yields the all-important motion of force. The D of this acceleration might be called second acceleration, but the notion has not yet proved useful in Mechanics (q.v.).
The notation for D may be this or that. Newton used the dot, thus s, to denote deriva tive as to t, as still do the British; Lagrange, the accent, F' (x), still common; Cauchy, the operator D, with or without subscribed argu ment x; others subscripts, as yx, etc.; most common, most expressive, but possibly mis dy leading is the Leibnitzian not a fraction (thus far at least), not the quotient of dy divided by dx, but the limit of the fraction dy for x vanishing, no matter how. Some dx times we write for Dz, thus: yz = d —dy etc.