Suppose there exist a function $f : \mathbb R \to \mathbb R$ such that $f(x)\cdot f'(x)$ is polynomial It. is trivial that if $f(x)$ is polynomial, then $f(x)\cdot f'(x)$ is polynomial.
My question is: how would one prove that $f(x)$ can be only polynomial?
It is not true in general that $f$ must be a polynomial, take for example $\,f(x) = \sqrt{x^2+1}\,$. What is true, however, is that $f^2$ must be a polynomial. Hint: $f \cdot f' = \frac{1}{2}\left(f^2\right)'$.