The derivative of a function measures how its output changes as its input changes. Formally, the derivative of f at a point x is the limit of the difference quotient as the change in x approaches zero.
f'(x) = lim (h -> 0) [f(x + h) - f(x)] / h
Integration is the reverse process: given a rate of change, integration recovers the total accumulated quantity. The Fundamental Theorem of Calculus links differentiation and integration, showing that one undoes the other.
Common derivative rules include the power rule, product rule, quotient rule, and chain rule. Each simplifies finding derivatives of more complex expressions built from simpler functions.