Derivatives. 171 



Divide both sides of (5) by J a- and we get 



~- = 2«r-f rJ;r. (6) 



Taking the limit of each side as J.r approaches zero and we get 



limit (Jjy] 



or T>^y=2ac. (8) 



Third. Find D,.r when y—cx^-\-eX'\-f, (i) 



lyCt x=^a and represent the corresponding value oi y by b and 



we get l=^ca''-\-ea-\-f. (2) 



Now let x=a-\- Ax and we get 



bJrM'^c(a^-Xxr^e(a-^Ax)-Vf, (3) 



or expanding and arranging, 



b-\-lv^cd'^ea-^f^-(2ac-\-e)Ax^-c(dx)\ (4) 



Subtract f 2J from (\) and we get 



Jy=^(2ac^e)Ax^c(Ax)\ (5) 



Divide both sides by J.r and we get 



Jy 

 '- — 2ac-\-c-\-cJX. (6) 



Taking the limit of each side as J.r approaches zero we get 

 limit \Ay\ , • , , 



or D..ji'=2^<r+<?. (8) 



12. In what precedes J.f has always been considered positive, 

 but J-^" may be negative, in which case x is increased by a nega- 

 tive quantity, or is really diminished, so it may be more proper to 

 call Ax the change in the zalne of x than to call it the amount by 

 which X has been increased. Either way of speaking is proper 

 provided we understand that the increase may be negative. In 

 any case Jx is the amount that must be added to one value of .r 

 to give another value of .r, and if the second value is greater than 

 the first the amount added will be negative. 



The same remark applies to Ay. 



