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We are tasked with finding the derivative of the given function by first expanding it. The function provided is:

\[
h(x) = \left( \frac{4}{x} + 7 \right)^2
\]

### Step 1: Expand the Expression

First, we expand the square:

\[
h(x) = \left( \frac{4}{x} + 7 \right)^2 = \left( \frac{4}{x} + 7 \right) \left( \frac{4}{x} + 7 \right)
\]

Using the distributive property (FOIL method):

\[
h(x) = \frac{4}{x} \cdot \frac{4}{x} + \frac{4}{x} \cdot 7 + 7 \cdot \frac{4}{x} + 7 \cdot 7
\]

\[
h(x) = \frac{16}{x^2} + \frac{28}{x} + \frac{28}{x} + 49
\]

Combine like terms:

\[
h(x) = \frac{16}{x^2} + \frac{56}{x} + 49
\]

### Step 2: Find the Derivative

Now that the expression is expanded, we can take the derivative \( h'(x) \).

\[
h'(x) = \frac{d}{dx} \left( \frac{16}{x^2} + \frac{56}{x} + 49 \right)
\]

- For \( \frac{16}{x^2} = 16x^{-2} \), the derivative is \( -32x^{-3} = -\frac{32}{x^3} \).
- For \( \frac{56}{x} = 56x^{-1} \), the derivative is \( -56x^{-2} = -\frac{56}{x^2} \).
- The derivative of the constant \( 49 \) is 0.

Thus, the derivative is:

\[
h'(x) = -\frac{32}{x^3} - \frac{56}{x^2}
\]

### Final Answer:

\[
h'(x) = -\frac{32}{x^3} - \frac{56}{x^2}
\]

Would you like more details on this process or have any specific questions?

Here are some related questions to deepen understanding:

1. What is the purpose of expanding the function before differentiating?
2. How does the chain rule compare to this method when differentiating powers of functions?
3. Can you differentiate the function without expanding it first?
4. How would the result change if the power was higher, say \( \left( \frac{4}{x} + 7 \right)^3 \)?
5. What are some practical applications of finding the derivative of such functions?

**Tip:** Always simplify the function as much as possible before applying differentiation rules to avoid mistakes in complex expressions.

Find the derivative of the following function by first expanding the expression. h(x) = (4/x + 7)^2

Find the derivative of the function h(x) = (4/x + 7)^2 by expanding the expression first. The solution involves polynomial expansion and applying derivative rules like the power rule and constant rule. This problem is suitable for students learning calculus at the college level.

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