Troubleshooting Newton-Raphson Iteration for Log-Likelihood Problem

[Kinetica]

Homework Statement

Hi guys!
I have solved a log-likelihood problem and its Newton-Raphson Iteration. My step-by-step solution is attached.

The only problem I have is when I plug in numbers, the final number for the first order Newton-Raphson is different from my professor's.

I guess you don't need any numbers. If you can can detect a mistake in my formula derivation, I will really appreciate it.

 

[vela]

Just to clarify, the t index variable of the summations is distinct from $\tau$, right? What depends on t? What happened to the term

$$\sum_{t=1}^n \log x!$$

?

 

[Kinetica]

Yep! It's different.
I am taking derivative with respect to that t-look-alike.
So I guess you treat x as a number.
But I cannot guarantee it.
Correct me if I am wrong please.

Just to clarify, the t index variable of the summations is distinct from $\tau$, right? What depends on t? What happened to the term

$$\sum_{t=1}^n \log x!$$

?