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We are given the function \( f(x) \), where \( f(1) = 10 \) and \( f'(1) = 5 \), and the function \( g(x) = x f(x) \). We are tasked with finding \( g'(1) \).

### Step-by-step solution:

1. **Differentiate \( g(x) \):**
   \[
   g(x) = x f(x)
   \]
   To differentiate \( g(x) \), we apply the **product rule**, which states:
   \[
   (uv)' = u'v + uv'
   \]
   In this case:
   \[
   u = x \quad \text{and} \quad v = f(x)
   \]
   So, applying the product rule:
   \[
   g'(x) = (x)' f(x) + x f'(x)
   \]
   This simplifies to:
   \[
   g'(x) = f(x) + x f'(x)
   \]

2. **Substitute \( x = 1 \):**
   Now, substitute \( x = 1 \) into the equation for \( g'(x) \):
   \[
   g'(1) = f(1) + 1 \cdot f'(1)
   \]
   Using the given values \( f(1) = 10 \) and \( f'(1) = 5 \), we get:
   \[
   g'(1) = 10 + 1 \cdot 5 = 10 + 5 = 15
   \]

Thus, the value of \( g'(1) \) is \( \boxed{15} \).

Would you like further details or have any questions? Here are some related questions for you:

1. What is the product rule for derivatives, and when is it used?
2. How would the solution change if \( g(x) = x^2 f(x) \) instead?
3. How can you verify the correctness of a derivative?
4. What would happen if \( f'(1) \) were 0 in this problem?
5. Can you explain the significance of evaluating functions at specific points?

**Tip:** Always remember to carefully apply the product rule when differentiating a product of functions!

The function f(x) is such that f(1) = 10 and its derivative f'(1) = 5. Given that g(x) = x f(x), what is the value of g'(1)?

Solve the calculus problem where f(1) = 10 and f'(1) = 5, and g(x) = x f(x). Learn step-by-step how to apply the product rule to find g'(1). This solution is ideal for students learning differentiation and the product rule.

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