Thread: Total Differentiation with Nested, Unspecified Composite Functions

I have no idea if the thread title accurately explains the question I'm having, but it is the best I could do. Googled and searched the forum and couldn't find what I was looking for.

Basically, I'm familiar with total differentiation. For instance I'm fairly certain that for a function

,

the total derivative is

.

But what if the composite functions are functions of many variables. For instance,

.

My first attempt for the total differential w.r.t. :



My second attempt:



Are either correct? Any help is appreciated!