DIFFERENTIAL AND INTEGRAL CALCULUS. 47 



or 



i, T\ n f \ ** 



since D^ operates on u only, 



D n ~ l D (uv] = -j- n _. . and so on, 

 ax ax 





w(n-l) rf'v cT"^ rf"v 



~[2T~ S 5 cfo"- "* ^ dx n ' 



Of course the same holds for the product of three or more 

 factors, the coefficients being those of the multinomial 

 theorem. 



d* (uvw) d*u d*v d*w 



TllUS dJ~ = VW dx* + WU dtf + UV d^ 



d*u dv du dv dw 



-j-s -7- + 5 terms of the same form + 6 -= r -^- , 

 aa? ie dx dx dx 1 



and in general 



Z^ t operating on u only, Z^ on v, Z> 3 on id?, but each 



d 



meaning -7- . 



The w tu differential coefficients of the functions sin#, 

 cos a*, logcc are readily found. 



Thus -j- (sin x] = cos x = sin (x + ^ TT) ; 



therefore 



= + ?r = COS + = 



