152 DIFFERENTIAL COEFFICIENT 



Proceeding to the limit, we obtain 



du fdu\ dv idu\ dy /du\ dz 

 dx \dv) dx \dyjdx \dz)dx' 



The process may be extended to the case in which u involves 

 more than three functions of x. 



175. Examples may occur more complicated in appear- 

 ance, but essentially involving the same principles as those 

 of the preceding Articles. Suppose for instance 



u = $(v, y, z, x), 



so that u could, by performing the requisite substitutions, be 

 made an explicit function of x : it is required to express the 

 differential coefficient of u with respect to x, without pre- 

 viously making these substitutions. 



du _ fdu\ dv /du\ dy /du\ dz (du 

 dx \dvj dx \dy) dx \dz) dx \dx 



4. . 



dx ~ \dy) dx \dz) dx \dx 



= - 



dx J v ' dx 



du fdu\ (fdv\ ., . . (dv\ , , . fdv\ 

 Hence -r- = M -p / (a;) + (--,- }F (as) + K- 

 dx \dvj \\dyj J \dz) \dxj 



176. The same suppositions being made as in Art. 169, 

 it is required to express -7-3 . We have 



du fdu\ dy fdu\ dz 

 dx ~ \dyj dx \dz J dx ' 



