Answer
\[\frac{-5}{2}\]
Work Step by Step
Given that $ h(2) = 4\;,\; h'(2)=-3$
Using quotient rule
$\frac{d}{dx}\left(\frac{h(x)}{x}\right)=\frac{h'(x)x-h(x).(x)'}{x^2}$
$\frac{d}{dx}\left(\frac{h(x)}{x}\right)=\frac{h'(x)x-h(x)}{x^2}$
$\left.\frac{d}{dx}\left(\frac{h(x)}{x}\right)\right|_{x=2}=\frac{h'(2)2-h(2)}{2^2}$
Using given data
$\left.\frac{d}{dx}\left(\frac{h(x)}{x}\right)\right|_{x=2}=\frac{(-3)(2)-4}{4}$
$\left.\frac{d}{dx}\left(\frac{h(x)}{x}\right)\right|_{x=2}=\frac{-6-4}{4}=\frac{-5}{2}$
Hence $\left.\frac{d}{dx}\left(\frac{h(x)}{x}\right)\right|_{x=2}=\frac{-5}{2}$.
