Let's analyze the problem step by step.

The task involves finding the derivative of the function $g(x) = x^2 - f(x) + 1$ at $x_0 = 2$, where $f(x)$ and its tangent at $x_0$ are shown in the graph.

Key Steps:

1. Understand $g'(x)$: $g(x) = x^2 - f(x) + 1$ Differentiating both sides with respect to $x$: $g'(x) = 2x - f'(x)$ So, at $x = 2$: $g'(2) = 2(2) - f'(2) = 4 - f'(2).$

2. Find $f'(2)$: From the graph, $f'(2)$ is the slope of the tangent line to $f(x)$ at $x_0 = 2$. The tangent line appears to rise by 1 unit for a run of 2 units (or similar ratios based on scaling).

   Thus: $f'(2) = \text{slope} = \frac{\Delta y}{\Delta x} = \frac{1}{2}.$

3. Calculate $g'(2)$: Substitute $f'(2) = \frac{1}{2}$ into $g'(2)$: $g'(2) = 4 - \frac{1}{2} = \frac{8}{2} - \frac{1}{2} = \frac{7}{2}.$

Final Answer:

$g'(2) = \frac{7}{2} \text{ or } 3.5.$

Let me know if you'd like a detailed explanation for any step! 😊

Related Questions:

1. How do you find the slope of a tangent line in general?
2. What happens if $g(x)$ includes more complex functions?
3. How do you apply the chain rule when differentiating $f(x)$?
4. What role does the tangent line play in interpreting derivatives?
5. Can you find higher-order derivatives for $g(x)$?

Tip: Always use the graph to double-check slopes visually for accuracy.