170 Algebra. 



10. The general method of finding the derivative ofy with re- 

 spect to X is that used in Art. 8, viz : give to x some value, say 

 a, and find the corresponding value of _r, then give to x a new 

 value, a + Jx, and again find the corresponding value of j/. Sub- 

 tract the first of the equations thus obtained from the second and 



we have the value of Jj. 



Jy 

 Divide both sides by Jx and we have the value of — . 



Jx 



Then finally find the limit of this fraction as Jjf approaches 



zero. 



11. We will now exemplify the method in a few examples. 

 First. Fi nd T^^y when y=/\.x^-\-^. ( i ) 

 I^et x=a and represent the corresponding value of j/ by b and 



we get <^=4«' + 5. (2) 



Now let x=a-\-Jx and the corresponding value of j' will be the 

 value d plus the amount by which j/ has been increased, or b-\-Jy, 

 hence d-\-Jy=4.(a + Jx/' + s. (3) 



Expanding, l?+ Jy=4.a^-\-8aJx-}-4( ~ix/-\-^. (4) 



Subtract (2) from (4) and we get 



J>/=8aXr-f4rJ-r/. . (5) 



Divide (^5 j by Jx and we obtain 



^=Sa-j-4(Jx). (6) 



Taking the limit of each side as Jx approaches zero we get 



limit (Jy) 



or D,j=8a. (8) 



Seco7id. Find Y)^y when y—cx^-\-e. (i) 

 Let x^=^a and represent the corresponding value of r by b, then 



b^^ca'-^e. (2) 

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



b^-Jy=.c(a^Jxy-^e, (3) 

 or expanding, 



b-\-Jy^ca^-\-2acJx-^c(JxyJre. (4) 

 Subtract (2) from (\) and we get 



Jy—2acJx-\-c(Jx)''. (5) 



