What is a derivative?

A derivative tells you how fast something is changing at any specific moment.

Simple idea: if a car travels 100 km in 2 hours, average speed = 50 km/h. But at some moments it was slower, others faster. A derivative gives the exact speed at any single instant, not just the average.

Formally, the derivative of a function f(x) measures the rate of change of f with respect to x. Geometrically, it gives you the slope of the curve at any single point.

Position, velocity, acceleration are connected by derivatives:

• Derivative of position → velocity (rate of change of position)
• Derivative of velocity → acceleration (rate of change of velocity)

Applications across fields:

• Physics: force is the derivative of momentum
• Economics: marginal cost is the derivative of total cost
• Biology: population growth rate is the derivative of population size