To compute the derivative of the function represented by this graph, we would follow these steps:

1. Identify Key Points on the Graph:

   • Look for the points where the slope of the tangent is 0, which correspond to the local maxima, minima, and points of inflection. These occur where the derivative of the function, $g'(x)$, is equal to 0.
   • You can also identify regions where the graph is increasing or decreasing to determine the sign of the derivative.

2. Estimate the Derivative in Different Intervals:

   • For $x < -3$, the graph appears to be increasing, so $g'(x) > 0$.
   • Near $x = -3$, there seems to be a local maximum, so $g'(x) = 0$.
   • Between $x = -3$ and $x = -1$, the graph is decreasing, so $g'(x) < 0$.
   • At $x = -1$, it appears to have a local minimum, meaning $g'(x) = 0$.
   • Between $x = -1$ and $x = 1$, the graph increases again, so $g'(x) > 0$.
   • At $x = 1$, another local maximum occurs, so $g'(x) = 0$.
   • Finally, for $x > 1$, the graph decreases, so $g'(x) < 0$.

3. General Observations:

   • The derivative is zero at critical points (local maxima and minima).
   • The derivative is positive when the function is increasing, and negative when the function is decreasing.

To get an explicit formula for the derivative $g'(x)$, you would need the equation for $g(x)$, but based on the graph, you can sketch the derivative by observing how the slope changes at different points.

Would you like further details on a specific part of this, such as how to estimate the derivative in each section?

Here are some follow-up questions for you:

1. Would you like to estimate the values of the derivative at specific points?
2. Are you interested in sketching the graph of the derivative $g'(x)$?
3. Do you need help finding the second derivative to discuss concavity?
4. Would you like to explore critical points in more detail?
5. Do you need to analyze the behavior of the function at infinity?

Tip: When interpreting a graph, always start by locating critical points, intervals of increase or decrease, and possible inflection points to determine where the function's derivative changes behavior.