Log likelihood and Maximum likelihood

[Casfgv]

I'm not sure how to get this first derivative (mainly where does the 4 come from?)
I know x̄ is the sample mean (which I think is 1/2?)
Can someone suggest where to start with finding the log-likelihood?

I know the mass function of a binomial distribution is:

 

[tkhunny]

You'll need a likelihood function representing the realization. [math]\left(\binom{1}{0}\cdot p^{0}\cdot(1-p)^{1}\right)\cdot\left(\binom{1}{1}\cdot p^{1}\cdot(1-p)^{0}\right)[/math], etc...

It may be helpful to note that [math]\overline{x} = \dfrac{1}{2}[/math]
Do you have the Method of Moments estimate?