Given a compact set $K\subset X_{\infty}^{{\rm reg}}$ we can find an integer $m(K)$ such that for $k\geq m(K)$, any point $x\in K$ and any tangent vector $v$ at $x$ there is a holomorphic section $s\in H^{0}(X_{\infty},L^{k})$ with $s(x)=0$ and the derivative of $s$ along $v$ not zero.