DIFFERENTIAL COEFFICIENTS. 43 



(3) If y = z v where both z and v are functions of x, then 

 by Art. 72 



fy _ g /^ ] 



finf* \ fi w 



CLJU \ilJs 



, v-dk\ 



H 7 - . 



z dx) 



74. Differential coefficient of x n . Third method. For 



the other methods see Arts. 47 and 48. 



The differential coefficient of x n is sometimes found thus : 

 First prove as in Art. 44 or 45 that if n be a positive 



integer, the differential coefficient of x n is wa;"" 1 . 



If then n be fractional and positive, suppose it = - where 

 p and q are positive integers. 







Let y= x n = x^, , 



therefore y q = x p . 



Hence taking the differential coefficients of both sides 



,7,/z-i fy. - ny*- 1 

 qy dx~ P ' 



dy _px p ~ 1 _p aT 1 



~~ ~ ' 



9. 

 The rule is thus established so long as n is positive. 



If n be negative suppose it = m, so that m is positive. 

 Let y = x, therefore 



/>. 



i 



y 



therefore 1 = yx m . 



Differentiate both sides, and we have 



= x m - + ymx m ~\ Arts. 29 and 33, 

 ax 



therefore 



