What Derivatives Are Used For in Practice

Speed, and then acceleration

The clearest answer to what derivatives are used for is motion. If a position is described by a function of time, its derivative is the velocity — the rate at which position is changing at that instant.

Differentiate again and you get acceleration, the rate at which the velocity is changing. A speedometer shows the first derivative; the push back into the seat is the second.

That last line is a physical fact recovered from a formula. Gravity being constant is exactly why the position is quadratic in the first place.

Finding the best value of something

At a maximum or a minimum, a smooth curve is momentarily flat, so its derivative is zero. That single observation is the basis of nearly all optimisation.

• Set the derivative equal to zero and solve — those are the stationary points.
• Use the second derivative to tell which is which: negative means a maximum, positive means a minimum.
• Check the endpoints too; the best value of a bounded problem is sometimes at an edge rather than at a turning point.

The applications are everywhere: the dimensions that enclose the most area for a given fence, the angle that throws a ball furthest, the production level that maximises profit, the shape that minimises material.

Marginal anything, in economics

Economists use derivatives constantly and call them marginal quantities. Marginal cost is the derivative of total cost — what one more unit costs to make.

The central result of the field follows immediately: profit is maximised where marginal cost equals marginal revenue, because that is where the derivative of profit is zero.

Rates of change in anything that is modelled

Once a quantity is described by a function, its derivative describes how quickly it moves, and that is usually the question worth asking.

Some of these have their own names and are taught as separate topics — current, velocity, growth rate — but they are one operation applied to different quantities.

Tangents, approximation and error

The derivative gives the slope of the tangent at a point, and near that point the tangent is a good stand-in for the curve. That is the basis of linear approximation: f(x + h) ≈ f(x) + h·f′(x).

It is more useful than it looks. It is how a calculator estimates a root by Newton's method, and how error propagates through a measurement: if a length is measured to within 1 mm and the area depends on it, the derivative tells you how far wrong the area could be.

Where it goes next

Differential equations are equations whose unknown is a function and which involve its derivatives. They describe cooling, radioactive decay, population growth, oscillation and the motion of essentially everything.

That is the real reason derivatives are worth learning properly. They are not a technique for answering exam questions about curves; they are the language in which physical change is written down.

Questions about what derivatives are used for