108 EXAMPLES OF INDEPENDENT DIFFERENTIATIONS. 



l'$7. If u be a function of the three independent variables, 

 x, y, z, we have in a similar manner 



d?u d'u 



dx dz dz dx ' 



d'u d*u 

 dx dy dy dx ' 



dx dy dz dx dz dy dz dx dy ' 

 and so on. 



EXAMPLES. 



T/ , xy , u . 



1. It u= * ,, rind -, r- and 

 ~ 



,, -, r- -, j- . 



a~ tr dx dy dy dz 



2. Verify in the following cases the equation 



dx dy dy dx ' 

 u = x sin y + y sin x, 

 u = xlogy, 



U = 3? } 



V 



u = log tan - , 



* 



_ ay bx 



U - 7 , 



by ax 



