\[\frac{\partial \hat{\psi}_p }{ \partial \hat{Z}}\]
\[\frac{\partial \hat{\psi}_p }{ \partial \hat{Z}}=
P_\psi \Bigg\{c_8 +c_9 \bar{R}^2 +2 c_3 \bar{Z}-8 c_4 \bar{R}^2 \bar{Z}+c_{11}
(3 \bar{R}^4 -12 \bar{R}^2 \bar{Z}^2)+c_6 (-24 \bar{R}^4 \bar{Z}+32 \bar{R}^2 \bar{Z}^3)
+c_{10} (3 \bar{Z}^2-3 \bar{R}^2
\ln{(\bar{R} )})+c_5 (-18 \bar{R}^2 \bar{Z}+8 \bar{Z}^3-24 \bar{R}^2 \bar{Z}
\ln{(\bar{R} )})
+c_{12} (-45 \bar{R}^4 +40 \bar{Z}^4+
60 \bar{R}^4 \ln{(\bar{R} )}-240 \bar{R}^2 \bar{Z}^2 \ln{(\bar{R} )})
+c_7 (150 \bar{R}^4 \bar{Z}-560 \bar{R}^2 \bar{Z}^3+48
\bar{Z}^5+360 \bar{R}^4 \bar{Z} \ln{(\bar{R} )}-480 \bar{R}^2 \bar{Z}^3 \ln{(\bar{R} )})\Bigg\} \]