I've been teaching calculus for several years and have some doubts about whether derivatives (and integration techniques) of common functions are useful and important outside mathematics and physics.
My question is:
Can you give an example of a natural problem outside mathematics and physics that can be solved using derivatives or integration, and cannot be solved simpler differently?
My motivation comes from trying to motivate students by good exercises.
The first constraint (naturality) excludes exercises such as "If $q$ units are produced in a factory, then your cost is $-0.3q^3+2q^2-\ldots$" and all of that kind, taught in microeconomics courses. The second constraint (cannot be solved simpler) exclude, for example, everything that leads to a minimum or maximum of quadratic expressions.
I'm aware of a few such instances, for example determining the length of the line segments in the solution of the Steiner tree problem for four points on a square (minimal road or electricity network connecting ABCD).