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We are tasked with finding \( h'(1) \) given that \( h(x) = x f(x) + 4 g(x) \), and we have the table of values for \( f(x) \), \( g(x) \), and their derivatives.

### Step 1: Differentiate \( h(x) \)
We start by differentiating \( h(x) = x f(x) + 4 g(x) \) using the product rule for \( x f(x) \):
\[
h'(x) = \frac{d}{dx} \left( x f(x) \right) + \frac{d}{dx} \left( 4 g(x) \right)
\]
Using the product rule on \( x f(x) \):
\[
\frac{d}{dx} \left( x f(x) \right) = f(x) + x f'(x)
\]
The derivative of \( 4 g(x) \) is:
\[
\frac{d}{dx} \left( 4 g(x) \right) = 4 g'(x)
\]
Thus, the derivative \( h'(x) \) is:
\[
h'(x) = f(x) + x f'(x) + 4 g'(x)
\]

### Step 2: Plug in \( x = 1 \)
We are asked to find \( h'(1) \), so we substitute \( x = 1 \) into the equation:
\[
h'(1) = f(1) + 1 \cdot f'(1) + 4 g'(1)
\]
From the table:
- \( f(1) = 3 \)
- \( f'(1) = -1 \)
- \( g'(1) = 4 \)

Now substitute these values into the equation:
\[
h'(1) = 3 + 1 \cdot (-1) + 4 \cdot 4
\]
\[
h'(1) = 3 - 1 + 16
\]
\[
h'(1) = 18
\]

### Final Answer:
\[
h'(1) = 18
\]

Would you like more details on any of these steps?

Here are five related questions to expand on this topic:
1. How does the product rule apply to functions like \( x f(x) \)?
2. What is the importance of knowing both the function and its derivative values from a table?
3. How would \( h'(x) \) change if \( h(x) = x^2 f(x) + 4 g(x) \)?
4. What is the chain rule, and how would it apply if the functions were compositions of other functions?
5. Can you explain how to find higher-order derivatives using similar methods?

**Tip:** When differentiating products of functions, always apply the product rule carefully and keep track of each term’s derivative.

Find h'(1) if h(x) = x f(x) + 4 g(x), using the values from the table for f(x), g(x), and their derivatives.

Learn how to differentiate the function h(x) = x f(x) + 4 g(x) using the product rule and calculate h'(1) with provided values for f(x) and g(x). This detailed solution involves applying the product rule and substitution of values from the table. Suitable for high school students studying calculus.

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