Figure 11

Use the graph of the function $y=$ $f(x)$, shown in Figure 11, to sketch the graph of the derivative function $y=f^\prime (x)$.

Solution We begin by drawing tangent lines to the graph of $f$ at the points shown in Figure 11. See the graph at the top of Figure 12. At the points $(-2, 3)$ and $\left(\dfrac{3}{2},-2\right)$ the tangent lines are horizontal, so their slopes are 0. This means $f^\prime ( -2) =0$ and $f^\prime \left(\dfrac{3}{2}\right) =0,$ so the points $(-2, 0)$ and $\left(\dfrac{3}{2},0\right)$ are on the graph of the derivative function. Now we estimate the slope of the tangent lines at the other selected points. For example, at the point $( -4,-3) .$ the slope of the tangent line is positive and the line is rather steep. We estimate the slope to be close to $6,$ and we plot the point $(-4,6)$ on the bottom graph of Figure 12. Continue the process and then connect the points with a smooth curve.