# Preservation of algebraically dependence for derivative [closed]

It is well-known (see Allouche monography for example) that if $f$ is an algebraic function over $K(X)$ then $f'$ is also algebraic. I wonder whether $f$ and $g$ are algebraically dependent, then $f'$ and $g'$ are also algebraically dependent. I think that it is false, but I do have no counterexample, neither a proof that this assertion is true.

• Every two algebraic functions of $x$ are evidently algebraically dependent. – Alexandre Eremenko Nov 25 '16 at 3:07
• @Alexandre, the hypothesis seems to be that $f$ and $g$ are algebraically dependent, not that $f$ and $g$ are algebraic. – Gerry Myerson Nov 25 '16 at 4:32
Example. Let $f=\Gamma(z)$, Euler's Gamma function, and $g(z)=\Gamma^2(z)$. Evidently they are algebraicaly dependent. Then $g'=2ff'$. Suppose that $f'$ and $g'$ are algebraically dependent, that is there is an equation $F(f',g')=0$ where $F$ is a polynomial with constant coefficients. Then $F(f',2ff')=0$ but this is an algebraic differential equation, and we know from Holder that $\Gamma$ does not satisfy any.