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We are tasked with finding the derivative of the expression \( \frac{h(x)}{x} \) evaluated at \( x = 2 \), given that \( h(2) = 2 \) and \( h'(2) = -6 \).

### Step-by-Step Solution:

We need to calculate:

\[
\left. \frac{d}{dx} \left( \frac{h(x)}{x} \right) \right|_{x=2}
\]

Using the quotient rule for differentiation, where:

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

Here, \( f(x) = h(x) \) and \( g(x) = x \), so we get:

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

Simplifying:

\[
\frac{d}{dx} \left( \frac{h(x)}{x} \right) = \frac{h'(x) \cdot x - h(x)}{x^2}
\]

### Now, substitute \( x = 2 \), \( h(2) = 2 \), and \( h'(2) = -6 \):

\[
\frac{d}{dx} \left( \frac{h(x)}{x} \right)\Bigg|_{x=2} = \frac{(-6) \cdot 2 - 2}{2^2}
\]

Simplifying further:

\[
= \frac{-12 - 2}{4} = \frac{-14}{4} = -\frac{7}{2}
\]

### Final Answer:

\[
\frac{d}{dx} \left( \frac{h(x)}{x} \right)\Bigg|_{x=2} = -\frac{7}{2}
\]

Would you like any further details or clarifications?

---

Here are 5 related questions to deepen your understanding:
1. How does the quotient rule apply to functions involving polynomials?
2. What is the physical interpretation of the derivative of a quotient in real-world problems?
3. How would the problem change if we were asked to differentiate \( \frac{h(x)}{x^2} \) instead?
4. Can you find the second derivative of \( \frac{h(x)}{x} \) at \( x = 2 \)?
5. What other rules of differentiation (e.g., product or chain rule) could be used to solve similar problems?

**Tip:** Always check if you can simplify the expression before applying the quotient rule, as it can make differentiation easier.

If h(2) = 2 and h'(2) = -6, find the derivative of h(x)/x evaluated at x=2.

This problem explores the application of the quotient rule to find the derivative of h(x)/x at x = 2, given h(2) = 2 and h'(2) = -6. It covers differentiation concepts for advanced high school or early university-level calculus.

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