\[ \frac{\partial \hat{\psi}_p }{ \partial \hat{R}} \]
\[ \frac{\partial \hat{\psi}_p }{ \partial \hat{R}} = P_\psi \Bigg\{ 2 c_2 \bar{R} +(\bar{R}^3 )/2+2 c_9 \bar{R} \bar{Z} +c_4 (4 \bar{R}^3 -8 \bar{R} \bar{Z}^2)+c_{11} (12 \bar{R}^3 \bar{Z}-8 \bar{R} \bar{Z}^3 +c_6 (6 \bar{R}^5 -48 \bar{R}^3 \bar{Z}^2+16 \bar{R} \bar{Z}^4)+c_3 (-\bar{R} -2 \bar{R} \ln{(\bar{R} )})+ A ((\bar{R} )/2-(\bar{R}^3 )/2+\bar{R} \ln{(\bar{R} )}) +c_{10} (-3 \bar{R} \bar{Z}-6 \bar{R} \bar{Z} \ln{(\bar{R} )})+c_5 (3 \bar{R}^3 -30 \bar{R} \bar{Z}^2+12 \bar{R}^3 \ln{(\bar{R} )}-24 \bar{R} \bar{Z}^2 \ln{(\bar{R} )}) +c_{12} (-120 \bar{R}^3 \bar{Z}-80 \bar{R} \bar{Z}^3+240 \bar{R}^3 \bar{Z} \ln{(\bar{R} )}-160 \bar{R} \bar{Z}^3 \ln{(\bar{R} )}) +c_7 (-15 \bar{R}^5 +480 \bar{R}^3 \bar{Z}^2-400 \bar{R} \bar{Z}^4-90 \bar{R}^5 \ln{(\bar{R} )}+720 \bar{R}^3 \bar{Z}^2 \ln{(\bar{R} )}-240 \bar{R} \bar{Z}^4 \ln{(\bar{R} )})\Bigg\} \]
with \( \bar R := \frac{ R}{R_0} \) and \(\bar Z := \frac{Z}{R_0}\)