Answer
$$ R'(2)= -\frac{57}{16}.$$
Work Step by Step
Given $ R(x)=f(x)/g(x)$, then we have $ R'(x)=\frac{g(x)f'(x)-f(x)g'(x)}{g^2(x)}$ and hence
$$ R'(2)=\frac{g(2)f'(2)-f(2)g'(2)}{g^2(2)}=\frac{4(-3)-5(9)}{4^2}\\
=\frac{-12-45}{16}=-\frac{57}{16}.$$
