176 Algebra. 



Let vw=v', \.\i&n y=tiv', and hence 



D.,J^/=z;'D,^^^-?^D,^;^ (2) 



But D^f ' = e£'D ,.u + vD.,zv, (t,) 



hence by substitution in (2) we have 



Now if we had any number of functions of x, say ti, r, u\ . . . 

 and if we let y be their product we have 



y=UVW2 ... (l) 



Let the product of all the functions after the first be represented 

 by a single letter, that is, let 



V'=^VW2 . . . 



then y=uv'. 



Find T>„-y as the product of two functions. Then 



B^y=i/I),,u-\-uB^v\ (2) 



Find D^t'' by letting v'=V7£'', where w' represents the product 

 W2 . . . 



Substitute the value thus found in (2). The result will con- 

 tain one temi involving T>_^.w'. 



Find the derivative by considering zv' to be the product of ^7i'o 

 factors. 



Continue this process until finally we reach the product of the 

 last two factors of the expression with which we started. 



The result may be stated thus : 



Tke derivative with respect to x of the product of any number of 

 functions is equal to the sum of all the prod^icts obtaijied by multi- 

 plying the derivative of each factor by the product of all the other 

 factors. 



If the equation here described be divided through by the prod- 

 uct of all the given functions, the result may be represented in 

 quite a convenient form, viz : 



-^^=-^4—^ + --^^ — h— ^-f- ... 

 y u V w 2 



22. Examples. 



Find the derivative with respect to x of the following expres- 

 sions without performing the multiplications indicated : 

 I. (x-{-i)(x-\-2) compare with example 4, Art. 19. 



