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We are tasked with finding the derivative of the function \( h(s) \) using the product rule, where:

\[
h(s) = \left( s^{-1/2} + 6s \right) \left( 7 - s^{-1} \right)
\]

The product rule states that if \( h(s) = f(s)g(s) \), then:

\[
h'(s) = f'(s)g(s) + f(s)g'(s)
\]

### Step 1: Define the functions \( f(s) \) and \( g(s) \)
We define:
\[
f(s) = s^{-1/2} + 6s
\]
\[
g(s) = 7 - s^{-1}
\]

### Step 2: Compute the derivatives
First, let's compute the derivatives \( f'(s) \) and \( g'(s) \).

1. **Derivative of \( f(s) \):**
   \[
   f(s) = s^{-1/2} + 6s
   \]
   Using the power rule, we get:
   \[
   f'(s) = -\frac{1}{2}s^{-3/2} + 6
   \]

2. **Derivative of \( g(s) \):**
   \[
   g(s) = 7 - s^{-1}
   \]
   The derivative is:
   \[
   g'(s) = s^{-2}
   \]

### Step 3: Apply the product rule
Now apply the product rule:
\[
h'(s) = f'(s)g(s) + f(s)g'(s)
\]
Substitute the values of \( f(s) \), \( g(s) \), \( f'(s) \), and \( g'(s) \):
\[
h'(s) = \left( -\frac{1}{2}s^{-3/2} + 6 \right)(7 - s^{-1}) + \left( s^{-1/2} + 6s \right)s^{-2}
\]

### Step 4: Evaluate at \( s = 16 \)
Now, substitute \( s = 16 \) into the expression for \( h'(s) \). Let me calculate this for you.The derivative of the function \( h(s) \) at \( s = 16 \) is approximately:

\[
h'(16) \approx 41.95
\]

Would you like further details or have any other questions?

Here are five related questions you might explore:
1. How does the product rule extend to functions with more factors?
2. What is the significance of the power rule when applied to fractional exponents?
3. How can we visualize the behavior of \( h(s) \) and its derivative?
4. How would the result change if the function had an additional multiplicative factor?
5. Can we use the quotient rule instead of the product rule in certain cases?

**Tip**: When differentiating functions involving powers of \( s \), make sure to apply the power rule carefully, especially with negative or fractional exponents.

Use the Product Rule to calculate the derivative for the function h(s) = (s^(-1/2) + 6s)(7 - s^(-1)) at s = 16.

Learn how to apply the Product Rule to differentiate the function h(s) = (s^(-1/2) + 6s)(7 - s^(-1)) at s = 16. This solution covers key concepts in calculus, including the Product Rule, Power Rule, and detailed step-by-step differentiation.

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