Let $f=f_0+(1-|x|^2)f_1+\cdots$ ($f_j$ harmonic) be a polyharmonic function of finite degree in the unit disc $B\in \mathbb R^2$. Let $X^\alpha=L^p(B,(1—|x|)^{p\alpha-1}dx)$, $0<p\leq\infty$, $\alpha>0$. It is proved that $\partial f/\partial x_l\in X^\alpha$ iff $|\operatorname{grad} f|\in X^\alpha$ iff $\partial f_j/\partial x_l\in X^{\alpha+j}$ for every $j$. There holds the analogous fact for higher order derivatives.