Property of homogeneous functions in two variables

Question

Why is $f_x(tx, ty) = t^{n-1}f_x(x, y)$ when $f(x, y)$ is a homogeneous function of degree $n$? What I came up with is that if $u = tx$, because $f(tx, ty) = t^{n}f(x, y)$, $$t^n\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u}\frac{\partial u}{\partial x} = \frac{\partial f}{\partial u}t$$ Am I getting it right?

Answer 1

Yes, you are correct. Congratulations!