CHAPTER III. 



DIFFERENTIAL COEFFICIENTS OF SIMPLE FUNCTIONS. 



44. Differential coefficient of x n where n is a positive 

 integer. 



Let y = x n , therefore 



y + &y=(x + h) n , 

 therefore Ay = (x + h) n - x n 



L* m 



therefore ^ = nx n ~* + n ( n ~ 1 ) 



Ace 1.2 



Diminish Jt without limit, and we have 



- 



1 IvJU 



ax 



45. The same result may also be obtained by means of 

 Art. 30. For let 



where the w quantities y x , y 2 , ... 2/ n , are all functions of a; ; 

 we have then 



1 du = !_ d& I dy^ ^ ^ 1 dy n 

 u dx y l dx 2/ 2 dx y n dx ' 



If now^ 2/ t = x, we have 



Ay t = Ax, 



therefore -r^- 1 = 1> 



A# 



therefore -r 1 = 1- 



