DIFFERENTIAL AND INTEGRAL CALCULUS. 51 



This may also be proved by the use of De Moivre's 

 Theorem, for 



_ 



-I)-* V(-1)4J 2 

 therefore 



fa*" 2 



_v(-^), 



2 



and if a; = cot 0, 



{x*J(- 1)} ?!+1 = (cos0 V(- 1) sin^}^ 1 - (si 



= (cos (n + 1) 9 V(- 1) sin (w + 1 ) 0)} -r (sin ^) n+1 ; 



d* +1 y Bin(n + 1) ^ (sin 2 ^ 1 



= -- i- 



= (- l) n [w sin (n + 1) ^ sin 7!+1 ^. 

 To differentiate x n loga?, take 



r, = na^* logo?4 *., . 



whence x -jj~= ny -f x n ; 



therefore 



therefore 



Now 



' n) 



