﻿Higher Derivative of a Function. 827 



d_ 

 du 



Let k successive operations u-t- each on <f)(u), that is 



[(" i)(" £)(*£) ■■■■" °p erations ] * w 



wherein w and -7- are not permutable, be designated by 



then 



( d \'*( \ % 1 vV n »/*V y K d K 4>(u) 



d\ 



Let M = ^,thenf«#V= «- =(Tf, 



therefore 



But (tf — a) » f « = £«*(# _ a y e (<-a)x 



_ a K d n e^~^ x 



dx n 



d n u K - a . 

 dx n 

 Hence 



III. A third way of proving the theorem is as follows : — 



dy dy du 



dx ~ du ' dx ' 



d 2 y d 2 u dy (du\ 2 d 2 y 

 a^ 2 = dP'dii + \Zv) die 2 ' 



but 

 therefore 



fdu\ 2 __l_rdW (2\ d 2 u\ 



\dx) -2l\_dx 2 ~\l) U dx 2 J y 



