174 Algebra. 



Divide both sides of (^) by Jx and we get 



Jx Jx'Jx *^^ 



Taking the Hniit of each side of (4.) as Jx approaches zero we get 

 Hmit j-^J^) limit f-^^) , limit \ Jv \ 

 -J^- :^ otj^|-J-r - o\-j^j-^Jx - oJXrj ^^^ 

 or D^r=D^2/ + D^z'. 



In the same way if y=z^ — z' we would get 



The result may be expressed thus : 



T/ie derivative of the algebraic sum of two fimctions of x equals 

 the algebraic sum of their separate derivatives. 



18. 'I'o FIND THE Derivative with Respect to x of the 

 Algebraic Sum of any Number of Functions of x. 



Let there be any number of functions of x represented by u, v, 

 7v, etc., and let their sum be represented by r; then we have 

 y=-u^v-\-2v-\- ... (i) 



Increase .t" by the amount Jx and suppose that u, v, it\ etc., 

 are increased by the amounts J?/, Jv, Jw, etc., respective!}* and j' 

 is increased by Jy, then we have after x is thus increased 



y-\-Jy=H-^Ju-i-v-\-Jv-\-2i.f-^J7i'-\- . . . (2) 



Subtract (i) from (2) and we get 



jy=ju-^jz,^j2c'-\- . . . (:,) 



Divide both sides o{ (2,) by Jx and we have 



Jy Ju Jv Jw 



i , 4-',-+ , + . . . (4) 



Jx Jx Jx Jx 



Taking the limit of both sides of (4) as Jx approaches zero we 



have 



D,r=D.,?^ + D,z' + D.,7£'+ . . . ^5^ 



If some of the signs in f i^ had been negative the same process 

 could have been applied and the result would have had negative 

 signs in the same positions as the}- appeared in the original 

 functions. 



The result may be expressed thus : 



77/^ derivative of the algebraic sum af several functions of x equals 

 the algebraic sum of their sepai'ate de?'ivatives. 



