Let $f: \mathbb{R} \to \mathbb{R}$ be a homogeneous function of degree 1 such that $\forall \lambda \in \mathbb{R}, \ \ f(\lambda x) = \lambda f(x)$. Let $y \in \mathbb{R}$
Can I do the following? \begin{align} \frac{d}{dx} y f(x) = \frac{d}{dx} f(yx) = \frac{d}{dx} f(y)x = f(y) \end{align}
In particular, can I obtain this result from Euler's theorem?
It seems correct to me. For example if $$ f(x)=kx $$ then $$ \frac{d}{dx}yf(x)=\frac{d}{dx}ykx=ky=f(y) $$ Euler theorem in this case can be expressed as $$ x\frac{d}{dx}f(x)=f(x) $$ and it can be applied to $yf(x),$ being itself homogeneous, so $$ x\frac{d}{dx}yf(x)=yf(x)=xf(y),\ \forall x\quad\implies\quad \frac{d}{dx}yf(x)=f(y) $$