To solve part of a problem I need to calculate $\frac{\partial}{\partial t} f(tx,ty)$. It's also given that $f(tx,ty)=t^n f(x,y)$ for some integer $n$. Yet I'm not exactly sure how to do this. I know I have to use the chain rule, instead of just evaluating $\frac{\partial}{\partial t} t^n f(x,y) = nt^{n-1} f(x,y)$. Since they will probably lead to different (but equivilent) expressions which can help me prove that $$ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}=nf(x,y) $$
My attempt consisted of letting $u=tx$ and $v=ty$. Then all I have to do is differentiate $\frac{\partial}{\partial t} f(u,v)$ which by the chain rule gives $$ \frac{\partial f}{\partial t} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial t} + \frac{\partial f}{\partial v} \frac{\partial v}{\partial t}=x\frac{\partial f}{\partial u} + y\frac{\partial f}{\partial v} $$
And now I'm stuck. It should be the case that $\frac{\partial f}{\partial u} = t^{n-1} \frac{\partial f}{\partial x}$ (and a similar thing for $y$) since these would lead me to the conclusion. At this point, I feel like I've done something wrong but I'm not sure how else to approach this.
Also, note that I don't want help with actually solving the problem, I just need a nudge in the right direction or a hint. And the hint I want is to evaluate $\frac{\partial}{\partial t} f(tx,ty)$.