According to textbook we are using at uni, the Euler's theorem which says that if $f(x,y)$ is homogeneous of of degree $k$ then:
$$x f_1'(x,y) + yf_2'(x,y) = kf(x,y)$$
textbook says that this can be proven by differentiating the following with respect to $t$:
$$f(tx,ty)=t^kf(x,y)$$
This should give us:
$$ x f_1'(tx,ty) + yf_2'(tx,ty) = kt^{k-1}f(x,y)$$
and setting $t=1$ gives us the equation of Euler theorem.
I understand why on right hand side we get $kt^{k-1}f(x,y)$, but I don't understand why chain rule gives us the left hand side expression. If we are differentiating with respect to $t$ why $f_1'$ and $f_2'$ which are derivatives with respect to $x$ and $y$ are there. It is not mentioned that $x$ or $y$ are functions of $t$, so I don't understand why it is not some $f_t'$.
Thanks for any help.