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