= Solution
Assume the <homogeneous function> is <differentiable>. Applying the <multivariable chain rule> to $f(tx,ty)=t^mf(x,y)$ gives, for $t$ near $1$,
$$
x f_x(tx,ty)+y f_y(tx,ty)=mt^{m-1}f(x,y).
$$
Setting $t=1$ proves <Euler theorem for homogeneous functions>:
$$
\boxed{x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=mf.}
$$
Only positive $t$ near $1$ are needed, so the derivation remains valid even when a noninteger $m$ makes the printed phrase “all real $t$” require qualification.
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