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For the complex quadratic family $q_c:z\mapsto z^2+c$, it is known that everypoint in the Julia set $J(q_c)$ moves holomorphically on $c$ except at theboundary points of the Mandelbrot set. In this note, we present short proofs ofthe following derivative estimates of the motions near the boundary points$1/4$ and $-2$: for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$is uniformly $O(1/\sqrt{1/4-c})$ when real $c\nearrow 1/4$; and is uniformly$O(1/\sqrt{-2-c})$ when real $c\nearrow -2$. These estimates of the derivativeimply Hausdorff convergence of the Julia set $J(q_c)$ when $c$ approaches theseboundary points. In particular, the Hausdorff distance between $J(q_c)$ with$0\le c<1/4$ and $J(q_{1/4})$ is exactly $\sqrt{1/4-c}$.

Simple Proofs for the Derivative Estimates of the Holomorphic Motion near Two Boundary Points of the Mandelbrot Set