$(2)(10$ points$)$ Let $r(t)$ be a function that measures the firing rate of a neuron at a given time $t.$ The units on $r(t)$ are spikes$/$second$,$ and the units on $t$ are seconds.

Suppose we know that the firing rate is $r(t)=5sin(\frac{t}{10})+3.$

$($a$)$ Use derivative rules to find a formula for $r^{\text{'}}(t).$

$($b$)$ Find all values of $t,$ within the range $0t10,$ such that $r^{\text{'}}(t)=0.$

$($c$)$ To find the global maximum of the function $r(t)$ over the interval $0t10,$ compare the values $r(0),r(10),$ and $r(t)$ for any critical point of the function. Identify the global maximum of $r(t),$ and describe what this means in terms of the real$-$world context of the problem.

$($d$)$ Calculate the second derivative $r^{\text{'}\text{'}}(t).$