SUCCESSIVE DIFFERENTIATION. 63 



Also since the theorem is supposed to hold for the value n 

 we have from (1), by changing v into -j- , 



<h<Fu_d?_( dv\_ dT^f d?v\ n(n-l) d* 

 dx dx n ~ dx"^ dx) n dx n - i ( U dx*)' { 1.2 da 



Now suppose the right-hand members of (2) and (3) written 

 so that the first term of (3) is immediately under the second 

 term of (2), the second term of (3) under the third term of (2), 

 and so on. Then by subtracting we have 



d n+1 u d n+1 uv 



v j-s+i = j ,ri-i (n + 1) ~r^ ( u IT } + ' , 

 dx ax dx \ ax/ 1 . 2 



d n+l v 

 dx n+1 



. . d n f dv\ (n+i}n d nl f dv 



(n + 1) -TIT ( u -T + , T~5I=i u TT5 

 J dx \ dx) 1 . 2 dx \ dx . 



- + (-l) n+l u 



This shews that if the theorem is true for a specific value 

 of n it is also true when n is changed into n+ 1. Therefore 

 since it is true when n = 1 it is universally true. 



EXAMPLES. 



d?y cos x 



1. If y = tan x + sec x, -^ = T - = r 3 . 



dx* (1 sm x) 



. 3 sin x sin 3i 



2. Let ^ = sin x = - - - , 



d"v 3 . / WTT\ 3" . 



3. 



4. 



K Tf / 2 , 2\ 4. -1 X 



5. If2/=(a ; 2 + a)tan 1 -, 



