I'm not sure what to do here with these given values. I can't find an example online or in my book similar to this. I have a feeling I'm probably supposed to use the chain rule, but I can't figure out how to apply it to this problem. Can someone please give me some guidance? Thanks.
The chain rule is exactly what you’re supposed to use here. It tells us that $$p'(t) = f_x(g(t),h(t))\,g'(t)+f_y(g(t),h(t))\,h'(t).$$ Plug in the known values: $$p'(2)=f_x(4,5)\,g'(2)+f_y(4,5)\,h'(2) = 2\cdot(-3)+8\cdot6=42.$$