Derivatives are incredibly useful for finding the highest point in an arc, the least cost for a procedure, or more generally the best way to do something. In fact, the word calculus comes from the Latin title of a paper by Leibniz:
A new method for maxima and minima as well as tangents, which is neither impeded by fractional nor irrational quantities, and a remarkable type of calculation (calculus) for them.
The basic idea is simple, although it takes some work to get the details right. If you throw a ball into the air, how fast is it rising when it reaches the top of the arc? The answer can't be positive, or else the ball would be even higher an instant later. The answer can't be negative, or else the ball would have been higher an instant earlier. So the answer must be zero. We find maxima and minima by looking for times when the rate of change is zero.