Owl: You've estimated a derivative at one time. Now imagine a weather balloon's altitude graph for an entire afternoon. At each moment where the path is smooth, there's a vertical velocity — up positive, down negative.
Collect those velocities into a second graph. That second graph is f': a function whose input is time and whose output is slope of altitude.
f' does not have to look like f. A hill-shaped altitude curve can produce a velocity that starts positive, crosses zero at the peak, then goes negative — more "downward slant" than "hill."
On the board: a rough altitude sketch. Without formulas, what should the velocity graph do while the balloon is rising steeply? While it's cresting? While it's descending?