Bayesian Variable Transformation (solve for PDF)

Hi I'm struggling with where to start for this variable transformation problem.

X~Bin(N,p) so that p(X=x) = (N choose x) * (p^x) * (1-p)^(N-x)

Use a uniform prior p(p)=1 for p in [0,1] (This is a Beta(1,1))

Then we'll determine the pdf for lambda, such that lambda is the log-odds, or

ln(p/(1-P))

Sooooooo

lambda = g(p) = ln(p/(1-p))

g inverse(lambda) = e^x/(e^x+1)

P(lambda) = p(g inverse(lambda)) * |d/dp*g inverse(lambda)|

p(p)=1 so

= |d/dp*g inverse(lambda)| = e^x/(e^2x +2e^x+1)

Is this right? I think the vast majority of this is right but I'm certain I've made one or two slip ups.