Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/26

16 INFINITESIMAL CALCULUS ( (7) If 2/ = e* r , prove that -- = e*C*(l +log a;). (8) If ?/=- 1 -, prove that ). Prove that 23. We shall conclude this section with the consideration of the differential of the area ABPM (fig. 3) of a plane curve, comprised between the curve, the axis of x, and two ordinates, of which one BA is fixed and the other PM is variable, x, y represent ing the coordinates of P. This area, when the equation of the curve is given, is an implicit function of x. If it be repre sented by u, we proceed to find its differential coefficient, or Suppose x to receive an in- . dx definitely small increment repre sented by MM , the correspond ing increment of the area is represented by PMM P , i.e., by the sum of the rectangle PMM R and the elementary area PP K. Now the latter area becomes evanescent in the limit in comparison with PMM R, du dx or - =<t> (x), where dx Consequently in proceeding to the limit we have l = PM = ?/, = <t>(x) is the equation of the curve. From this we can make an important inference, viz., that in all cases there exists a function whose differential coefficient is any given function of x, suppose <t>(x). To find such a function it is sufficient to consider the curve whose equation in rectangular coordinates is ?/ = <(.i ) ; then the area comprised between any fixed ordinate and the ordinate whose abscissa is a; is a determinate function,- which, by the preceding, has <p(x) for its derived function. Successive Differentiation. 24. We have seen that from any function of a variable we can obtain by differentiation a new function, called its differential co efficient, or, after Lagrange, its derived function. If the primitive function be represented by/(x), then, as already stated, its first derived function is denoted by f(x). If this new function, f(x), be treated in the same manner, its derived function is called the second derived of the original function /(a;), and is denoted by f"(x). In like manner, the derived function of f (x) is the third derived of f(x), and represented by f"(x), &c. In accordance with this notation, the successive derived functions of f(x) are represented by each of which is the derived function of the preceding. 25. In like manner, if y=f(x), then d jl *=f(x) . Hence dy dx) df(_x) dy dx) , The function - dx is written | , and is called the second differential coefficient of?/ with regard to x. Likewise is written and so on ; dx do? and the series of functions dy d?y_ d dx dx* dx are called the first, second, third, the function represented by y. It is sometimes convenient to adopt a notation analogous to that of fluxions, and to represent the series of differential coefficients of d>y dx" nih differential coefficients of y , y" , y " , 2/">, in order to abbreviate the labour of writing down the system of successive differential coefficients. 26. It is plain that the determination of the series of successive derived functions of any function of x does not require any new principles, as it is accomplished by successive applications of the methods already considered. For example, if y--=x", we have dy dx nj Again, if y=--c ax , we have dii d 2 i/ y = rw* c , -~!, = a 2 c ax , &c., dx il.c- and in general t ~=a n c,< ui . dx" 27. We next proceed to a fundamental theorem due to Leibnitz, and first published in Mis. Bcrol., 1710, viz., to find the nth de rived function of the product of two functions. Let y = uv; then, if we write y , u , v , y", it", &c. , for dy du dv d z y dx dx dx dx" we have y = uv + vu . The next differentiation gives y" = uv" + u v + v u + vu" = u v" + 2u v + vu". The third differentiation gives y " = uv " + u v" + lu v" + 2u"v + v u" + vu " = uv " + 3u v" + 3u"v + vu ", in which the coefficients are the same as those in the expansion of (a + b) s . Suppose that the same law holds for the nth differential coefficient, and that I . then, differentiating again, we get in which the coefficients follow the law of the Binomial Expansion. Accordingly, if this law hold for any integer value of n, it holds for the next higher integer ; but it holds wiien w = 3, therefore it holds for 7i = 4, &c. In the ordinary notation the preceding result is written dx" 1 . 2" (1) If T/ = C sin bx, to find J? . dx n Here ~J. = e ax (a sin bx + b cos bx). dx Now let b = a tan <p, and we have - ^=( 2 + & 2 )i c ax (sin bx cos + cos bx sin </>) dx = (a 2 + & 2 )* e ax sin (bx + <p). Similarly we get -^| = (a 2 + b 2 ) e ax sin (bx + 20) ; and, in general, ^= ft s + j/^r c ax s j n ( (I) If ?/- cot- l x, to find . " Here hence dy = sin 2 v/ ( sin 2 //) = sin 2 // sin2?y . dy =- (s^V Bin2y)-^ ~ (sin 2 ?/ siuty) dx dx dij = sin 2 ?/ c - (sin 2 y sin2?/) dy -1.2 sin 3 ?/ sin3?/. In like manner, ^=1 . 2 . 3 sin- 1 ?/ sin 4y. And, in general, C H = ( - 1)" . -l sin"?/ siimy. (Ex. 4, 22.