Implicit differentiation; reproducing textbook derivation

[D_Tr]

Homework Statement

(The fourth equation is the central one)

first, we have \frac{1}{r}=\frac{a}{b^2}(1+ecosθ) and b^2=a^2(1-e^2)

now using these two, we transform

acosψ=ae+rcosθ into (1-ecosψ)(1+ecosθ)=\frac{b^2}{a^2}

we want to find dθ/dψ, and the author performs an inplicit differentiation, and the result in the book is (treating θ as a function of ψ)
dθ/dψ=\frac{b}{a(1-ecosψ)}

Homework Equations

stated above

The Attempt at a Solution

I performed the implicit differentiation and got:
esinψ(1+ecosθ)-esinθ\frac{dθ}{dψ}(1-ecosψ)=0

Is my implicit differentiation wrong or are some transformations needed? I tried to match the result in the book using the first two equations together with my result without success..

 

[D_Tr]

Never mind, I found it. I just needed to use a relation between sinψ and sinθ (which I have not posted here).