I'm studying Liouville's theory of elementary integration and I came across an unproven fact that the derivative of an algebraic function is algebraic. I can see this intuitively, but I can't rigorously prove it using the algebraic function definition.
A function y is algebraic if it satisfies an equation of the form
$$P_ny^n+P_{n-1}y^{n-1}+\cdots+P_1y+P_0=0,$$ where each $P_1,\ldots,P_n$ is a polynomial.
I tried to differentiate this equation and found
$$P'_ny^n+nP_ny^{n-1}y'+P'_{n-1}y^{n-1}+(n-1)P_{n-1}y^{n-2}y'+\cdots+P'_1y+P_1y'+P_0'=0,$$ but I can't manipulate this equation to find an algebraic equation involving just $y'$. If anyone has a tip or solution for this I'd appreciate it.