We are tasked with finding $\frac{d}{dx} \left( \frac{h(x)}{x} \right)$ evaluated at $x = 2$, given that:

$h(2) = 3 \quad \text{and} \quad h'(2) = -2.$

To approach this, we will use the quotient rule for differentiation. The quotient rule states that for two differentiable functions $f(x)$ and $g(x)$:

$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{g(x) f'(x) - f(x) g'(x)}{(g(x))^2}.$

In our case, $f(x) = h(x)$ and $g(x) = x$. Let's apply the quotient rule:

$\frac{d}{dx} \left( \frac{h(x)}{x} \right) = \frac{x \cdot h'(x) - h(x) \cdot 1}{x^2}.$

Now, evaluate this expression at $x = 2$:

$\frac{d}{dx} \left( \frac{h(x)}{x} \right)\bigg|_{x = 2} = \frac{2 \cdot h'(2) - h(2) \cdot 1}{2^2}.$

Substitute the known values $h(2) = 3$ and $h'(2) = -2$:

= \frac{-4 - 3}{4} = \frac{-7}{4}.$$ Thus, the value of $$\frac{d}{dx} \left( \frac{h(x)}{x} \right)$$ at $$x = 2$$ is: $$\boxed{-\frac{7}{4}}.$$ Would you like more details or have any further questions? --- Here are 5 related questions to expand your understanding: 1. How do you differentiate the product of two functions using the product rule? 2. What is the derivative of $$\frac{1}{x}$$, and how does it relate to the quotient rule? 3. If $$h(x)$$ were a specific polynomial, how would that change the derivative calculation? 4. How would the quotient rule change if $$g(x)$$ were more complex than just $$x$$? 5. What are some practical applications of the quotient rule in physics or economics? **Tip:** Always check if you can simplify the quotient before applying the quotient rule. This can sometimes make differentiation easier.