[R] Analytical Optimization (Stat question)

[francogrex]

Dear Experts, this is more a general stat question, I tried to ask in other
places but had no luck with answers (expect one that suggested numerical
instead of analytical optimization):
The likelihood below is a mixture of two negative binomial 
distributions: 
P*f(n;x1,x2,E) + (1-P)*f(n;x3,x4,E) 
N and E are vectors of same length. I would like to find the fist 
derivatives with respect to x1,x2,x3,x4 and P (so that I can set them 
to zero and calculate a bayesian MLE by iterations according to  David
Draper's 
"Bayesian Modeling, Inference and Prediction"). 
Does anyone have a "quick and dirty" way? To avoid drawning in rather 
complicated math... 
Here's the transcription of the likelihood in S language: 
P*exp((lgamma(x1+N)))-(((lgamma(x1)+lfactorial(N))+log((1+(E/ 
x2))^x1)+log((1+(x2/E))^N)))+ 
(1-P)*exp((lgamma(x3+N)))-(((lgamma(x3)+lfactorial(N))+log((1+(E/ 
x4))^x3)+log((1+(x4/E))^N))) 
-- 
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