To find the first derivative of the function $g(x)$, represented by the graph in the image, we need to analyze the graph's slopes at different points, as the derivative represents the slope of the tangent line at each point on the curve.

Visual Observations of $g(x)$:

1. Intervals of increase:

   • The curve is increasing (positive slope) from $x \approx -6$ to $x \approx -2$.
   • It also increases after $x = 5$.

2. Intervals of decrease:

   • The curve is decreasing (negative slope) from $x \approx -2$ to $x = 2$.

3. Critical points (where the slope is zero):

   • At $x \approx -2$, there is a local maximum.
   • At $x \approx 2$, there is a local minimum.
   • At $x = 5$, there seems to be a point of inflection, where the concavity changes.

Behavior of the First Derivative $g'(x)$:

1. Negative Slope: Between $x = -2$ and $x = 2$, the curve has a negative slope, so $g'(x) < 0$ on this interval.
2. Zero Slope: The derivative is zero at $x = -2$ and $x = 2$ (where the local maximum and minimum occur), indicating horizontal tangents.
3. Positive Slope: The slope is positive for $x < -2$ and $x > 5$, meaning $g'(x) > 0$ on those intervals.
4. Inflection Point: At $x = 5$, the curve changes concavity, meaning $g'(x)$ changes but is still positive on either side, suggesting a slope near zero but not zero itself.

If you need further details, I can guide you through sketching the derivative or any additional analysis!

Follow-up questions:

1. How can you identify the critical points on the graph of a function?
2. What is the significance of the inflection point in terms of the second derivative?
3. How can we estimate the derivative at a particular point using the graph?
4. What happens to the derivative as $x \to \infty$ in this graph?
5. How would the first derivative graph look based on this curve?

Tip:

When analyzing a graph, remember that critical points occur where the slope of the tangent is zero, which corresponds to local maxima, minima, or saddle points.