MLE - 1st and 2nd derivative of the function

Hi guys again, it seems I haven't been doing the maximum likelihood
estimation correctly. I quote below, can someone explain to me please
what does it mean that the 2nd and 3rd derivatives of the function
equals zero and how to compute that in R (or any other program but I
have R). Thanks.

We have our initial estimated, subjective parameters for the gamma
mixture and we have our likelihood that is the mixture of negative
binomials representing the distribution of actual observed values. We
'pool' these distributions and determine which expression for the
parameters would be most likely to produce the sample of observed
negative binomial counts (determine the MLE). This maximisation
involves a search in five-dimensional parameter space {θ: α1,α2,
β1, β2, P} for the vector that maximises the likelihood as evidenced
by the first and second derivatives of the function being zero. The
likelihood is
L(θ) = Πij {P f (Nij; α1, β1, Eij) + (1-P) f (Nij; α2, β2, Eij)}
This involves millions of calculations. The computational procedures
required for these calculations are based on the Newton-Raphson method.
This is an old calculus-based technique that was devised to find the
roots of an equation (e.g. the values of the independent variable (e.g.
x) for which the value of the function (e.g. f(x)) equals zero.
To summarise, maximum likelihood estimation involves the following four
steps: (i) obtain your data; (ii) specify a model; (iii) compute the
likelihoods and; (iv) find the value of the parameters that maximises
the likelihood.

.

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