draw the derivative for the following graphs. 386.

Explanation:

Step1: Recall derivative - slope relationship

The derivative of a function at a point is the slope of the tangent line to the function at that point.

Step2: Analyze intervals of increasing and decreasing

• When the function is increasing, the derivative is positive. In the given graph, the function is increasing on some sub - intervals of ((-3,3)).
• When the function is decreasing, the derivative is negative. There are sub - intervals where the function is decreasing in ((-3,3)).

Step3: Identify critical points

Critical points occur where the derivative is zero (horizontal tangent). In the graph, there are points where the tangent is horizontal. At these points, the value of the derivative is 0.

Step4: Sketch the derivative graph

Start by marking the x - values of the critical points on the x - axis of the derivative graph. Then, for intervals of increase, draw the derivative graph above the x - axis, and for intervals of decrease, draw it below the x - axis. The height of the derivative graph represents the magnitude of the slope of the original function.

Since this is a drawing task and not a numerical calculation, a full - fledged numerical answer cannot be provided in a traditional sense. The key steps to draw the derivative graph are as described above. The derivative graph will cross the x - axis at the x - values corresponding to the critical points of the original function, be above the x - axis for intervals where the original function is increasing and below the x - axis for intervals where the original function is decreasing.