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!