Sketching Graphs of Derivative Functions

Previously, we have seen that if \( f(x) \) is a polynomial of degree \( n \), then its derivative is one degree lower (i.e., \( n - 1 \)). (One exception to this is the case where \(f(x)\) is a constant function and so has degree \(n=0\).)

For example, if \( f(x) \) is a cubic polynomial (degree \(3\)), then its derivative function is quadratic (degree \(2\)). The derivative of the quadratic function is linear, so the second derivative function of a cubic polynomial is linear (degree \(1\)).

Also, as we move from the left side to the right side of the graph of a polynomial with degree \(n\geq 2\), we notice that the slope of the tangent line changes in steepness and over certain intervals the slope is positive or negative.

The slope of the tangent line changes from positive to negative as we pass through a local maximum.

The slope of the tangent line changes from negative to positive as we pass through a local minimum.

At the local maximum or minimum, we see that the tangent line is horizontal and therefore, the value of its derivative is \(0\).

We can use all of this information to help us sketch the graph of the derivative function.