Let

${H}_{\alpha}^{\infty}$ and

${A}^{p}\left(\varphi \right)$ denote the spaces of analytic functions in the unit disk such that

${sup}_{\left|z\right|<1}{(1-|z|}^{2}{)}^{\alpha}\left|f\left(z\right)\right|<\infty $ and

${\int}_{D}{\left|f\left(z\right)\right|}^{p}\frac{{\varphi}^{p}\left(\right|z\left|\right)}{1-\left|z\right|}\phantom{\rule{0.166667em}{0ex}}dA\left(z\right)<\infty $ for a normal weight function

$\varphi $, respectively. The author completely characterizes the boundedness and compactness of the composition between differentiation and multiplication operator

$D{M}_{u}$ and also for the operators

${D}_{\phi ,u}^{n}f=u{f}^{\left(n\right)}\circ \phi $, where

$u$ is a given analytic function and

$\phi $ is a non-constant analytic self-map on the disk, and between the spaces

${A}^{p}\left(\varphi \right)$ and

${H}_{\alpha}^{\infty}$. Due to the fact that

${D}_{\phi ,u}^{n}$ gives

$D{C}_{\phi}$,

${C}_{\phi}D$ and

${M}_{u}D$ as particular cases, his results allow him to unify a number of previously known theorems.