SUCCESSIVE DIFFERENTIATION. 61 



Now the series (3) follows the same law as (1). Hence 

 if for any value of n the formula in (1) is true, it is true 

 also for the next greater value of n. But we have proved 

 that it holds when n = 3 ; therefore it holds when n = 4, 

 therefore when n=5, and so on ; that is, it is universally true. 



This theorem is called after the name of its discoverer, 

 Leibnitz. 



81. If u = e M cos bx ; we have by Arts. 78 and 80, 



r\ n(n-l] n _ 2 , 8 / 2?r\ 



i/ fcJ br < ^H* - *TJ 



We may also find another form for this w* differential 

 coefficient as follows: 



- - = e ax (a cos bx b sin bx) ; 



assume a = r cos <f>, 



b = r sin <, 

 so that r =( a + &)*, 



thus T- = re ax cos (bx + <), 



where r and <f> are constant quantities. 



fZ 2 M 



Similarly j-j = re** (a cos (fee + $ - b sin (fa? + <)} 



= rV* cos (fa; 

 and generally 



cf e o* cos i x 



cos 



82. The following is an important example of Art. 80. 

 Let u = 



d> n e ax 

 then, remembering that -j-ir cfe *, we have 



