Rates of Change in Finance and Science

The derivative was defined as a limit of a difference quotient; its economic interpretation is the core of marginal analysis. Marginal cost, marginal revenue, and marginal profit are the derivatives of the corresponding total functions — the instantaneous rate of change of cost, revenue, and profit with respect to output. Every field that models a quantity as a function of another uses the derivative as its central sensitivity: velocity in physics, growth rate in biology, rate of reaction in chemistry, current in electronics. This note collects and illustrates these interpretations, with the goal of reading the derivative fluently across contexts and recognising that one abstract object — $dy/dx$ — unifies them all.

• interpret $dy/dx$ as an instantaneous rate of change in any applied context;
• compute velocity, speed, and acceleration from a position function;
• define and compute marginal cost, marginal revenue, and marginal profit, and interpret them as approximate next-unit costs;
• apply the derivative to population growth rates and reaction rates.

1. The unified framework
2. Physics: position, velocity, and acceleration
3. Biology: population growth rate
4. Chemistry: rate of reaction
5. Economics: marginal analysis
6. A single idea, many forms