The derivative measures the instantaneous rate of change of a function with respect to its variable.
Geometrically, the derivative f'(a) equals the slope of the tangent line to the curve y = f(x) at x = a. The formal definition is the limit of the difference quotient: f'(x) = limh→0 [f(x+h) - f(x)] / h. A function is differentiable at x if this limit exists. Differentiability implies continuity, but continuity does not guarantee differentiability (e.g., f(x) = |x| is continuous but not differentiable at x = 0). Key differentiation rules: the power rule (ddx[xⁿ] = nxⁿ⁻¹), product rule ((uv)' = u'v + uv'), quotient rule ((u/v)' = (u'v - uv')/v²), and chain rule (ddx[f(g(x))] = f'(g(x))·g'(x)). Derivatives are used in physics (velocity v = ds/dt, acceleration a = dv/dt = d²s/dt²), economics (marginal cost = dC/dq, marginal revenue = dR/dq), optimization (critical points where f'(x) = 0), machine learning (gradient descent for minimizing loss functions), and curve sketching (increasing/decreasing intervals, concavity via second derivative). Higher-order derivatives: f''(x) measures concavity (curvature direction), f'''(x) measures jerk in physics. Partial derivatives extend the concept to multivariable functions: ∂f/∂x measures change with respect to x while holding other variables constant.
The derivative concept emerged in the 17th century from two problems: finding tangents to curves (geometry) and computing instantaneous velocity (physics). Pierre de Fermat (1630s) developed a method for finding tangents using "adequality," essentially the limit concept. Isaac Newton (1665) developed the "method of fluxions," calling derivatives "fluxions" (rates of flow) and variables "fluents." Newton denoted fluxions with a dot: ẋ. Gottfried Leibniz (1684) introduced the superior notation dy/dx and provided systematic rules. The notation f'(x) was introduced by Joseph-Louis Lagrange (1797). Augustin-Louis Cauchy (1823) provided the first rigorous limit-based definition. Karl Weierstrass later developed the ε-δ formulation still taught today. In the 20th century, the concept was extended to distributions (Laurent Schwartz, 1945) and Fréchet derivatives in infinite-dimensional spaces.