Let $f(x)=x^2-1$

$($a$)$ Find the average rate of change of $f$ over the interval $-1,1.$

$($b$)$ Find the derivative of $f$ with respect to $x$ using the limit definition.

$($c$)$ Find the slope of the line tangent to $f$ at the point $(4,15).$

$($d$)$ Find the instantaneous rate of change of $f$ when $x=-3$