62 SUCCESSIVE DIFFERENTIATION. 



If now the expression 



d\ n 

 -j- y 



dxj 9 



be expanded by the Binomial Theorem, and the symbols 

 fd\ /d\* fd\ 3 



(&)* Us/* yy*~ 



replaced by 



dy d*y d'y 



'3?' ^respectively, 



the result will be the same as the series in parentheses in (1). 

 Hence, we may write 



as a convenient abbreviated method of stating the equation (1). 



83. The following theorem is sometimes of use in the 

 higher branches of mathematics. 



If n be any positive integer 



d n u_d n uv d n ~ l ( dv\ n(n-l] d n ~* ( d 2 v 

 ~~~x)^ 1.2 & 



This theorem may be readily established by Induction. 

 For it is obviously true when n = l, and if we assume it to 

 be true for a specific value of n we can shew that it will be 

 true when n is changed into n+ 1. Assume that (1) is true 

 and differentiate both sides ; thus 



d n+1 u dv d n u _ d n+l uv cP_/ dv\ 

 n ~ n * " n W 



_ 

 dx dx n ~ dx n * " dtf V fo/ 1.2 



