28 DIFFERENTIAL COEFFICIENT OF A POWER, 



i 



Suppose v=z r , then 



z r I if* I v p v q 



or 



z-l z-\ v r -l v q (v r -l) 



v q (v r -l) 



This last result is obtained by dividing both numerator and 

 denominator of the preceding fraction by v 1. Let now v 

 approach the limit 1, then the limit of the last fraction is 



therefore = -= a?"' 1 = nx n ~\ 



ax r 



48. Differential coefficient of x n . Second method. 

 Let y = x n , therefore 



. Ay (x + h} n -x n 



therefore -r 1 = - - r 



Ax h 



Assume - = z and (1 + ^)" l = v, then z and v are quantities 



M? 



which diminish indefinitely with h. Thus 



Aa; 



From the above assumptions 



therefore log, (1 + v) = n log e (l + 2). 



