We are given that $G(x) = 4f(x) - g(x)$, and we need to find $G'(2)$, which is the derivative of $G(x)$ evaluated at $x = 2$.

Using basic differentiation rules, we know:

$G'(x) = 4f'(x) - g'(x)$

Thus, to find $G'(2)$, we need to compute $4f'(2) - g'(2)$.

Steps:

1. From the graph, we will estimate the slopes (derivatives) $f'(x)$ and $g'(x)$ at $x = 2$.
2. We will multiply $f'(2)$ by 4, then subtract $g'(2)$ from it to get $G'(2)$.

Looking at the graph:

• For $f'(2)$: The graph of $f(x)$ appears to be a straight line at $x = 2$, and the slope is the change in $y$ over the change in $x$. By estimation from the graph, we see that for $f(x)$, the slope appears to be negative. A rough estimate gives $f'(2) \approx -1$.

• For $g'(2)$: Similarly, for $g(x)$, the slope at $x = 2$ also looks negative but steeper than $f(x)$. A rough estimate gives $g'(2) \approx -2$.

Now substitute these estimates into the equation for $G'(2)$:

$G'(2) = 4(-1) - (-2) = -4 + 2 = -2$

So, $G'(2) \approx -2$.

Would you like more details on how to estimate slopes from graphs?

Here are some related questions:

1. How is the derivative of a function defined geometrically?
2. How do you differentiate a function with multiple terms like $G(x) = 4f(x) - g(x)$?
3. What is the significance of the slope of a function at a particular point?
4. How do you compute the derivative when dealing with piecewise or non-linear graphs?
5. Can we use numerical methods to estimate derivatives from graphical data?

Tip: When estimating slopes from graphs, focus on identifying two clear points on the line and use the formula for the slope: $\text{slope} = \frac{\Delta y}{\Delta x}$.