Help with using the multivariable chain rule

I need to show how the multivariable chain rule can be derived to show the product rule for differentiation. I can see how the product rule is in the multivariable chain rule but can't mathematically show it. Any help appreciated.

Let f(x,y) be a function of two variables, and let g(t) and h(t) each be functions of one variable. Then the multivariate chain rule says that if

F(t) := f(g(t), h(t)),

then

dF/dt = df/dx dg/dt + df/dy dh/dt,
where we evaluate df/dx and df/dy at x = g(t), y = h(t). (*)

Now let f(x,y) := xy. Then df/dx = y and df/dy = x, so according to (*),

dF/dt = y dg/dt + x dh/dt
= h(t) dg/dt + g(t) dh/dt.

This is the product rule for differentiation.