Differentiating the Tolman-Oppenheimer-Volkoff (TOV) pressure equation

[S.H.2013]

Homework Statement

I need to differentiate the pressure P(r) equation directly below, for the case where the density is constant (i.e. ρ(r) = ρ_c), to show that it is of the same form as the dP/dr equation further below:

Homework Equations

P(r) = rho_c×c²×( (√(1-2βr²/R²) - √(1-2β))/(√(1-2β) - √(1-2βr²/R²)) )

β = GM/Rc²

dP/dr = -G[( (ρ(r) + P(r)/c²)×(m(r) + 4∏r^3P(r)/c²) )/ r(r - 2Gm(r)/c²)]

dm/dr = 4r²ρ(r)

m(r) = M(r/R)³

ρ(r) = ρ_c

The Attempt at a Solution

I tried to differentiate the P(r) using the quotient rule (combined with chain rule) and substituted in for β and m(r) but got stuck at a point and don't know what to do next.

The point that I got stuck at is when:

dP/dr = 2GMrc²/R³×{ [3(1-2GM/Rc²)^1/2/(1-2GMr²/c²R³)^1/2 - 1] - [1 - (1-2GM/Rc²)^1/2/(1-2GMr²/R³c²)^1/2 ] } / { [10-18GM/Rc²-2GMr²/R³c²-6(1-2GM/Rc²)^1/2(1-2GMr²/R³c²)^1/2] }

 

[lippyka]

I'm not sure if this is the correct form or if I should just keep going with differentiating further.