Derivative Illustration

This plot illustrates the relationship of the derivative of a function to the function itself. The function f is shown as a thick blue curve. Its derivative is shown as a thin blue line. The tangent line to f is shown as a red line. The point where it is tangent is marked with a small diamond. The slope of the tangent line at that value of x is the value of the derivative at that point - at the end of the short gray line segment joining f with its derivative. You can see clearly that whenever the derivative function is negative, then the tangent line slopes downhill toward the right. When the derivative is positive, the tangent line slopes upward.

Derivative Plot
Point x0 where tangent is shown:


  1. Find the points where the slope of the tangent line is zero. What does \(f\) do at those points?
  2. Find the region where \(f\) is decreasing. How did you find it?