I'm not sure how to differentiate the log-likelihood with respect to beta j, so I can find the MLE?
Can anyone help me?
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Unfortunately we can't. We don't know the distribution of each $\beta_j$ so we can't answer your question. It looks like your beta's are probably from a logistic regression model, but I don't like to make assumptions. I think large chunks are missing from your question to allow anyone to answer this without assumptions. I'd recommend editing with all the necessary details of the question. Also add the self-study tag and tell us what you've tried. – StatsStudent May 31 '19 at 15:40
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@StatsStudent I read this question as asking how to differentiate the given expression with respect to $\beta_j,$ as illustrated at https://stats.stackexchange.com/a/37640/919. This requires no more information than that given (and is solved using the rules of differentiation, needing no more than the derivatives of the log and exp functions along with the Chain Rule). Could you explain your interpretation, which seems radically different? – whuber May 31 '19 at 16:50
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I read through the lines, the corresponding notation, (and "so I can find the MLE") as "can you help me solve this maximum likelihood problem." Of course, you are right that that if it's simply a differentiation problem, it's easily solved. But from context, it seemed to me to be a "help me find the MLE" question. – StatsStudent May 31 '19 at 18:13