Applications of derivatives outside mathematics and physics

Question

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).

Answer 1

There are plenty of differential equations in biology. The list from Wikipedia is:

• Verhulst equation – biological population growth

• von Bertalanffy model – biological individual growth

• Lotka–Volterra equations – biological population dynamics

• Replicator dynamics – found in theoretical biology

• Hodgkin–Huxley model – neural action potentials

And there are also ones coming from pharmacokinetics.

Of course you don't actually have to use the real differential equations coming from these fields, you could choose the mathematics you want to ask about (for example, a PDE) and then just say

The amounts of Drug X and Drug Y in a person's system satisfy the following system of PDE...

Answer 2

Applications of derivatives outside mathematics and physics

Would you consider finance and economics as part of $($applied$)$ mathematics ? Or did you have only “pure” mathematics in mind when you wrote this ? Anyway, the point is that all economic concepts whose name starts with the word “marginal” are basically the derivative or variation of the notion in question.

Answer 3

I think that given the way you have set up the problem, the answer could very well be negative. Because we could always talk about the quotient $$\frac{f(x+h)-f(x)}{h}$$ for an $h$ that is small enough, which is a simpler object than the derivative.