I have the function $$f(tx,ty)$$ and I want to take the partial derivative of this with respect to $t$. So set $x'=xt$ and $y'=yt$. I applied chain rule and got
$$\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x'}\frac{\partial x'}{\partial t} +\frac{\partial f}{\partial y'}\frac{\partial y'}{\partial t}$$
which yields
$$\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x'}x +\frac{\partial f}{\partial y'}y$$
But how do I evaluate that? It is still in terms of $x'$ and $y'$. How do I get it in terms of partials of $x$ and $y$?