Logarithmic derivatives of algebraic functions and differential algebra

I will begin with a brief discussion of a fundamental result in complex analysis known as the Lemma on the Logarithmic Derivative and its applications to the study of meromorphic solutions to algebraic differential equations.

In basic p-adic and non-Archimedean analogs of complex analysis, the lemma on the logarithmic derivative is trivial.

Let $\mathbf{C}(t)$ be the field of rational functions with complex coefficients. An element of a finite field extension of $\mathbf{C}(t)$ is called an algebraic function. In the complex case, the Lemma on the Logarithmic Derivative extends to algebraic (and more generally algebroid) functions. However, this remains open in p-adic and non-Archimedean cases. This could likely be resolved if basic questions about the irreducible polynomials of derivatives were known. More precisely, let $P(X)\in\mathbf{C}(t)[X]$ be a non-linear irreducible univariate polynomial with coefficients in $\mathbf{C}(t).$ Let $f$ be an algebraic function such that $P(f)=0.$ Let $f'$ be the $t$-derivative of $f,$ which is also an algebraic function. What can be said about the irreducible polynomial of $f'?$