Predators handling prey

In $1959,$ C$.$S$.$ Holling proposed the following model for the number of prey $P$ eaten by a predator during a fixed time period:

$P(a,N,T)=\frac{aN}{1+\text{a}\text{T}\text{N}}$

where $a>0$ is the attack rate, $N$ is the prey density, and $T$ is the time spent catching, moving, eating and digesting the prey.

$($a$)$ Calculate the partial derivatives $\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{a}},\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{N}}$ and $\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}.($Feel free to use software to evaluate them or to check your work.$)$

$($b$)$ Interpret the derivatives. Are they positive or negative? What do you conclude? Do these conclusions seem reasonable?

$($c$)$ Now compare how a unit increase in a versus a unit decrease in $T$ will impact the number of prey eaten. What is $P$ more sensitive to: a change in $a$ or a change in $T?$ Answer this biological interpretation by using the results of your calculations in part $($a$).$