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).