DIFFERENTIAL COEFFICIENT OF A POWER. 27 



Put then 2/1 > 2/2 > #> all equal to x; thus u becomes x n , 

 and we obtain 



1 du _n 



udx x' 



therefore -7- = nx n ~ l . 



ax 



46. If n be not a positive integer, we may by assuming 

 the truth of the Binomial Theorem for fractional exponents 



dx n 

 proceed as in Art. 44 to determine -,- . But in that case we 



shall require to assume that " if we have a series containing 

 an infinite number of terms and each term becomes ulti- 

 mately indefinitely small, the sum of the terms becomes so 

 too." To avoid this assumption we adopt another mode. 



47. Differential coefficient of x n the exponent n being un- 

 restricted. 



Let 2/ = a? n , therefore 

 therefore 



'x + _ 



h 



Now whatever be the value of n, positive or negative, whole 



nr\ _ fl 



or fractional, it may be supposed = *- - , where p, %, r, are 

 positive integers. 



T *** l~ ** 



Let - = z, 



x 



4" ri OTOT/YPO ' n 'y f<y 1 1 



UlclcIUlO It */ \Z ~ ij, 



^y ~ n 1 



and -7 = 



Aa; z 1 ' 



As h diminishes indefinitely z approaches the limit 1, and we 



z n 1 



have to find in that case the limit of . 



z 1 



