from sympy import pi, var, S, init_printing, log, Function, sqrt
init_printing(use_latex=True)

var("n")
ex = -3/(4*pi) * (3*pi**2*n)**(S(1)/3)

Vx = (n*ex).diff(n)
Vx

Vx/ex

rs = (3/(4*pi*n))**(S(1)/3)
rs

var("A B C D")
ec = A*log(rs) + B + C*rs*log(rs) + D*rs
ec

Vc = (n*ec).diff(n).expand()
Vc

rs.diff(n) / rs

rs = Function("rs")(n)
ec = A*log(rs) + B + C*rs*log(rs) + D*rs
ec

Vc = (n*ec).diff(n).expand()
Vc

Vc = Vc.subs(rs.diff(n), (-1/(3*n))*rs)
Vc

var("gamma beta1 beta2")
ec = gamma / (1+beta1*sqrt(rs)+beta2*rs)
ec

Vc = (n*ec).diff(n)
Vc

Vc = Vc.subs(rs.diff(n), (-1/(3*n))*rs)
Vc

C = (Vc / ec).simplify()
C

C.normal()