Design a box that has to hold exactly one litre, and there's no single correct shape — a tall thin box and a short wide one can both hold the same volume, using dramatically different amounts of cardboard, aluminium, or glass to make the walls. Somewhere among the infinite possible shapes is the one that uses the least material for that fixed volume. Finding it doesn't require building and measuring a single prototype. It requires setting a derivative equal to zero.
Turning a shape problem into a function to minimise
The first step in any optimisation problem like this is expressing what you're trying to minimise (surface area, in this case) as a function of a single variable, using the constraint (fixed volume) to eliminate the other variables. For a box, you might express the height in terms of the volume and the base dimensions, substitute that back into the surface area formula, and end up with a single equation describing surface area purely as a function of one dimension. That function's graph, plotted out, would show surface area rising toward infinity at extreme shapes (very tall and thin, or very short and wide) and dipping down to some minimum somewhere in between — and it's that dip you're hunting for.
Where the slope goes flat is where the minimum lives
A function's derivative describes its slope at every point, and at a smooth minimum — the very bottom of that dip — the slope is momentarily flat, neither rising nor falling: the derivative equals exactly zero there. This is the core insight optimisation calculus exploits: rather than checking every possible shape one at a time, you take the derivative of the surface-area function, set it equal to zero, and solve. Whatever dimension satisfies that equation is a candidate for the minimum (a second check confirms it's genuinely a minimum rather than a maximum or a saddle point), and that single algebraic solution instantly identifies the most material-efficient shape out of infinitely many possibilities, all without ever physically testing one.
What we're still unsure about
The mathematics of using derivatives to find minima and maxima is completely settled calculus, applied routinely and reliably across engineering, economics, and design. What isn't a matter of calculus at all, but of practical judgement, is deciding which real-world quantity is actually worth optimising in the first place — a mathematically minimal-material box might be an awkward shape to manufacture, stack, or ship, and real design decisions have to weigh the pure optimisation answer against constraints calculus alone doesn't capture, which is exactly why "the calculus-optimal answer" and "the answer a company actually builds" often turn out to be two different shapes.
This sits inside Applications of Derivatives, one of eight topics in Calculus, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.