If $ f$ is homogeneous of degree $ n$ , $ f(tx,ty) = t^{n}f(x,y)$

show that $ f_{x}(tx,ty) = t^{n-1}f_{x}(x,y)$

My proof went a little wrong as follow:

$ u=tx, v=ty \quad f_{x}(tx,ty) = \frac{\partial f(u,v)}{\partial u} \cdot \frac{\partial u}{\partial x}+\frac{\partial f(u,v)}{\partial v}\cdot \frac{\partial v}{\partial x} = f_{u}(u,v) \cdot t$ …….(1)

$ \quad\quad\quad\quad\quad\quad\frac{\partial}{\partial x}(t^nf(x,y))=t^nf_{x}(x,y)$ …….(2)

(1)=(2)$ \qquad f_{u}(u,v)=t^{n-1}f_{x}(x,y)$

$ \quad \quad \quad\quad \,f_{u}(tx,ty) = t^{n-1}f_{x}(x,y)$

On the last line, the footnote on the left side is supposed to be $ x$ , however, I get $ u$ .