Derivatives. 169 



8. In general, if a be taken as the value of x from which we be- 

 gin to count the increase of jt we would judge from analogy that the 



Jy 

 fraction —- approaches 2a as Jx approaches zero, or, using the 



notation of Chapter XI, Art. 22, 



hmit (Jy) 



This we wall now prove. 



Since y=x^-\-i, (i) 



whatever value be assigned to x the equation will enable us to 

 compute the corresponding value of y. 



First, let x=^a and represent the corresponding value oi y by 

 b, then b=a'-\-\. (2) 



Now let x=a-\-Jx 



and represent the corresponding value of_y by b-\-Jy, then from 

 equation (i) we get 



d-hJy=(a-hJxr-\-i, (3) 



or simplifying, 



d-\-Jy=a'-\-2aJx+(JxJ'+i. (4) 



Subtract (2) from (4) and we get 



Jy^2aJx-\-(Jx)\ (5) 



Divide (s) by Jx and we obtain 



:^=2a-\-Jx. (6) 



Jx 



As Jx varies, of course the the two sides of equation (6) are 

 variables, and, indeed, they are two variables that are always 

 equal, and as Jx approaches zero these two variables each ap- 

 proach a limit. 



Hence b}- Chapter XI, Art. 7, their limits must be equal. 



Therefore j^^l"^^ 



2a. 



Jy 

 Q, Definition. The value of the fraction -^ when that frac- 



Jx 



. Jy 

 tion is constant, or the limit of the fraction -~ as Jx approaches 



zero when that fraction is a variable, is called the Derivative of y 



tvith respect to x, and is represented by the notation D,j', where j' 



is a function of x. 

 A— 21 



