Question: Show that exists infinitely linear transformations $D \colon \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfies $D(fg) = D(f)g + f D(g)$ for all $f,g \in \mathbb{R}[x]$ in which $D$ is the derivative over the vector space of all polynomial.
I know the definition od linear transformation, and I think to use the contradiction to prove this problem. But I not sure that my opinion is correct.
For every constant $c$ the linear transformation $f \to cf'$ satisfies the given identity. So there are infinitely many of them.