any help with these questions would be much appreciated, thank you.
Assume that the response variable $Y$ in a regression problem is a Bernoulli random variable, that is $Y$ ~ $Be(\pi(\beta'x))$, where $\pi(\beta'x)$ is a logistic function, $\beta'x=\beta_0+\beta_1x_1+\beta_2x_2+...+\beta_px_p$ and $x=(1,x_1,x_2,...,x_p)$, i.e., $Y$ follows a logistic regression.
Let $(x_1,y_1),(x_2,y_2),...,(x_n,y_n)$ be a data set of independent samples, where $y_i$ ∈ {$0,1$} and $x_i=(1,x_{i1},x_{i2},...,x_{ip}), i=1,2,...,n$.
a) Show that for all real $\beta$, the log likelihood function $l(\beta)$ can be written as
$l(\beta)=∑y_i\beta'x_i-∑ln(1+exp(\beta'x))$.
b) Find the partial derivatives $∂/∂β_0$ $l(\beta)$ and $∂^2/∂β^2_0$ $l(\beta)$ in the form they would appear in a recursive algorithm like Newton-Raphson for finding the maximum likelihood estimate.