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YCor
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Does there exist another form of derivate for polynomials?

Consider linear $F: \mathbb R[x] \rightarrow \mathbb R[x]$ with : $F(P\times Q)=H(F(P),F(Q),P,Q)$ for some $H$.

A possible solution for $H$ is $H(u,x,y,z)=u\times z+x \times y$, where $F$ is the classical derivate.

Does there exist any other non-trivial solutions for $H$ ?

Dattier
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