  Let $\mathcal{F}\subset\mathcal{M}(D)$ and let $a, b$ and $c$ be three
distinct complex numbers. If, there exist a holomorphic function $h$ on $D$ and
a positive constant $\rho$ such that for each $f\in\mathcal{F},$ $f$ and
$f^{'}$ partially share three pairs of functions $(a,h), \ (b, c_f)$ and
$(c,d_f)$ on $D,$ where $c_f$ and $d_f$ are some values in some punctured disk
$D^*_{\rho}(0),$ then $\mathcal{F}$ is normal in $D$. This is an improvement of
Schwick's result[Arch. Math. (Basel), \textbf{59} (1992), 50-54]. We also
obtain several normality criteria which significantly improve the existing
results and examples are given to establish the sharpness of results.
