Suppose that $Y_{1}, \ldots, Y_{n}$ are independent random variables, and that $Y_{i}$ has probability density function

$$f\left(y_{i} \mid \theta_{i}, \phi\right)=\exp \left[\frac{\left(y_{i} \theta_{i}-b\left(\theta_{i}\right)\right)}{\phi}+c\left(y_{i}, \phi\right)\right]$$

Assume that $\mathbb{E}\left(Y_{i}\right)=\mu_{i}$ and that there is a known link function $g(.)$ such that

$$g\left(\mu_{i}\right)=\beta^{T} x_{i}$$

where $x_{1}, \ldots, x_{n}$ are known $p$-dimensional vectors and $\beta$ is an unknown $p$-dimensional parameter. Show that $\mathbb{E}\left(Y_{i}\right)=b^{\prime}\left(\theta_{i}\right)$ and that, if $\ell(\beta, \phi)$ is the log-likelihood function from the observations $\left(y_{1}, \ldots, y_{n}\right)$, then

$$\frac{\partial \ell(\beta, \phi)}{\partial \beta}=\sum_{1}^{n} \frac{\left(y_{i}-\mu_{i}\right) x_{i}}{g^{\prime}\left(\mu_{i}\right) V_{i}}$$

where $V_{i}$ is to be defined.