Physics: velocity and acceleration
The original application — the one derivatives were invented for. If s(t) gives an object’s position at time t, then its derivative s′(t) is the velocity (how fast position is changing) and the second derivative s″(t) is the acceleration (how fast velocity is changing).
A concrete example: throw a ball straight up at 20 m/s. Physics gives its height as s(t) = −4.9t² + 20t. Differentiate with the basic rules:
v(t) = s′(t) = −9.8t + 20
a(t) = s″(t) = −9.8
Reading the results: the velocity starts at 20 m/s and loses 9.8 m/s every second — the constant acceleration of gravity, which is exactly what the second derivative reports. The ball reaches its peak when the velocity hits zero: −9.8t + 20 = 0, so t ≈ 2.04 seconds. That move — set the derivative to zero to find the turning point — is the single most used trick in applied calculus, and it reappears below.
Economics: marginal cost and revenue
In economics, “marginal” is a code word for “derivative.” If C(x) is the total cost of producing x units, the marginal cost C′(x) tells you approximately what the next unit will cost to produce. Suppose a factory’s cost function is C(x) = 0.01x² + 5x + 2000 (in dollars). Then C′(x) = 0.02x + 5: at 100 units, the next unit costs about $7; at 1,000 units, about $25. Costs are accelerating — useful information when deciding whether to scale up.
Businesses run the same analysis on revenue and profit. A company maximizes profit where marginal revenue equals marginal cost — which is the economist’s way of saying “where the derivative of profit is zero.”
Optimization: finding the best
Maximums and minimums live where the derivative equals zero — at a peak or a valley, the tangent line is flat. That turns “find the best X” problems into differentiation problems.
The classic: you have 100 meters of fence and want the rectangular enclosure with the biggest area. With width x, the length is 50 − x (half the fence goes to each pair of sides), so the area is:
A(x) = x(50 − x) = 50x − x²
A′(x) = 50 − 2x
50 − 2x = 0 → x = 25
The best enclosure is the 25 × 25 square, with 625 m². The same set-the-derivative-to-zero logic designs shipping boxes that minimize cardboard, antennas that maximize signal, and routes that minimize fuel — optimization is probably the most economically valuable thing derivatives do.
Medicine and biology: rates of growth and decay
When a drug is administered, its concentration in the bloodstream rises, peaks, and decays — and the derivative of the concentration curve tells clinicians how fast. The peak (derivative zero again) determines when the drug is most effective; the decay rate determines how often doses are needed.
Population biology leans on the same tool: exponential growth models like P(t) = P₀e^(kt) are built directly on the fact that eˣ is its own derivative — the defining property of a population whose growth rate is proportional to its current size (covered in the transcendental functions guide).
Engineering and everyday life
Engineers use derivatives to find where stress on a beam is greatest (the maximum of the stress function), how quickly a circuit responds to a change in voltage, and how steeply a road can curve safely at a given speed. Weather models, GPS navigation, and machine learning training loops all run on derivatives under the hood — gradient descent, the algorithm that trains neural networks, is nothing but “follow the derivative downhill.”
And the most familiar example was in the intro guide all along: your car’s speedometer is a real-time derivative of your position, and the accelerator pedal is named after the second one.
Try the math yourself
Every model in this guide was differentiated with the same handful of rules from the earlier guides — nothing more exotic than the power rule and the exponential rule. Take any of the functions above — the ball’s height -4.9t^2 + 20t, the cost curve, the fence area — and run them through the derivative calculator to see the steps worked in full. (Tip: set the variable to t for the physics ones.)