Assume you observe data $X_t$ for $t=t_1,\dots,t_N$ where each data point is Poisson distributed but with a different parameter, i.e. $X_t \sim Pois(\lambda_t)$ for all $t \in \lbrace t_1,\dots,t_N\rbrace$.
Assume further that $\lambda_t = a*t+b$, i.e. the parameter changes linearly over time.
Now I am trying to find the maximum likelihood estimator (MLE) for $\lambda_t$, i.e. I need to find the MLE for $a$ and $b$.
I think the log likelihood function looks as follows:
$$\ell(\lambda_t,x_1,\dots,x_N) = \sum_{t=1}^N(-\lambda_t) -\sum_{t=1}^N \ln(x_t!) + \sum_{t=1}^Nx_t\ln(\lambda_t)\\ = \dfrac{-a*N*(N+1)}{2} -N*b- \sum_{t=1}^N \ln(x_t!) + \sum_{t=1}^Nx_t\ln(a*t+b)$$
And I belive the derivative in regards of $a$ can be formulated as
$$ \dfrac{d\ell}{d a} = -\dfrac{N(N+1)}{2} + \sum_{t=1}^Nx_t\dfrac{1}{a*t+b}t$$
And now I am stuck as I don't know how to, after setting this equal to $0$, simplify it further to solve for $a$. Basically the same happens if you take the derivative in respect to $b$.
I have never been good at this kind of thing, maybe someone can point me in the right direction or tell me if I made a mistake.