David:

Your confusion is really just a matter of notation, I think. Think of it this way:

The function f depends on each of the variables x_1... x_n in some way. By the notation

\partial f / \partial x_i


we mean to denote "however f responds to a small change in whatever is in its x_i-th component." If we multiply that by however much what is in that x_i-th component changes when t changes, we have the effect of t on f via its influence on whatever is in that x_i-th component. Adding these up for all the components x_i gives us the total effect t has had on f through all the ways it indirectly affects it. Hence, P.2 gives a correct application of the chain rule.

To see this maybe more clearly, take a simple case. For example, in the case of a function of one variable, say y=f(x), if in turn x=g(t), then we would write

dy/dt = (df(g(t))/dx )*dg(t)/dt...right?

We would not write (df/dg)(dg/dt) because strictly speaking f does not depend on g--it depends on x -- it is then x that depends on t and that dependence is captured in the second term.

Does this help?