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.


Any hints or answers will be welcome.
Thanks in advance



Preservation of algebraically dependence for derivative
The answer is no.


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.


