Question

if (h(2)=9) and (h(2)= - 2), find (left.\frac{d}{dx}left(\frac{h(x)}{x}
ight)
ight|_{x = 2}).

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = h(x)$ and $v=x$. So, $\frac{d}{dx}(\frac{h(x)}{x})=\frac{h'(x)\cdot x - h(x)\cdot1}{x^{2}}$.

Step2: Substitute $x = 2$

We know that $h(2)=9$ and $h'(2)=-2$. Substitute these values into the derivative formula: $\frac{h'(2)\cdot2 - h(2)\cdot1}{2^{2}}$.

Step3: Calculate the result

$\frac{(-2)\times2-9\times1}{4}=\frac{-4 - 9}{4}=\frac{-13}{4}$.

Answer:

$-\frac{13}{4}$