Thanks for your reply. But I am still puzzled.

By "the x_i-th component", do you mean "x_i", or, in this case, "x_i*t"? If the latter, it kind of makes sense, but it seems strange.

Another question is that in your explanation, you say "we have the effect of t on f". But it seems to me that this effect would be described as df/dt, not df/dx.

Am I right in thinking that this:
d
- (f(tx)) is just another way of writing:
dx

df(tx)
------ ?
dx

If so, then the line in the book just above P.2 seems to say that df(tx)/dx = itself times dtx/dx (forgetting the i's for ease of typing). This is very puzzling to me.

In the spirit of your closing example, of one variable, here is how I see this (which I know has to be wrong, but I still don't see my mistake).

let u = g(x) = tx
let y = f(g(x)) = f(u)
then dy/dx = (dy/du)*(du/dx)
but du/dx = d(tx)/dx = t
so dy/dx = (dy/du) * t
= (dy/dtx) * t
= (df(g(x))/dtx) * t
= (df(tx)/d(tx) * t

So I still end up with d(tx) in the denominator of P.2 rather than dx.