The derivative tells you how fast something is changing. For a function f(x), the derivative f′(x) gives the slope of the tangent line at each point. If f is position, the derivative is velocity. If f is velocity, the derivative is acceleration.

Formally, the derivative is a limit: the slope of a secant line as the two points get closer together. Newton called it a fluxion; Leibniz wrote dy/dx. Their notation and ideas became the foundation of differential calculus.

Derivatives are used to find maximum and minimum values, to approximate functions, and to model rates of change in physics, biology, economics, and engineering. Rules like the product rule, chain rule, and quotient rule make differentiation systematic. Partial derivatives extend the idea to functions of several variables. The derivative is one of the most powerful tools in mathematics.

• Measures rate of change of a function
• Slope of the tangent line
• Defined as a limit
• Rules include product, chain, and quotient