Differentiation is the process of finding the derivative of a function, measuring how it changes.
Differentiation is the systematic application of rules to compute derivatives. The fundamental differentiation rules are: (1) Power rule: ddx[xⁿ] = nxⁿ⁻¹ for any real exponent n. (2) Constant rule: ddx[c] = 0. (3) Constant multiple rule: ddx[cf(x)] = cf'(x). (4) Sum/difference rule: ddx[f ± g] = f' ± g'. (5) Product rule: (fg)' = f'g + fg' — "first times derivative of second plus second times derivative of first." (6) Quotient rule: (f/g)' = (f'g - fg')/g² — "low d-high minus high d-low over low squared." (7) Chain rule: ddx[f(g(x))] = f'(g(x))·g'(x) — "derivative of outer times derivative of inner." For trigonometric functions: ddx[sin x] = cos x, ddx[cos x] = -sin x, ddx[tan x] = sec² x. For exponential and logarithmic functions: ddx[eˣ] = eˣ, ddx[aˣ] = aˣ ln a, ddx[ln x] = 1/x, ddx[logₐ x] = 1/(x ln a). Implicit differentiation handles equations like x² + y² = r² without solving for y explicitly: differentiate both sides and solve for dy/dx. Logarithmic differentiation takes logs before differentiating, useful for products of many factors. Parametric differentiation: if x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt). Higher-order derivatives: f''(x) (second derivative) measures concavity, f'''(x) (third derivative) is jerk in physics, fⁿ(x) denotes the nth derivative. Applications: finding maxima/minima (f'(x) = 0), Newton's method for root-finding, related rates problems, differential equations, and linearization (tangent line approximation).
Differentiation rules developed alongside calculus. Newton's "method of fluxions" (1660s) used what he called "prime and ultimate ratios." Leibniz published the first systematic account of differentiation rules in his 1684 paper "Nova Methodus pro Maximis et Minimis." The chain rule was explicitly stated by Leibniz in 1676. Guillaume de l'Hôpital published the first calculus textbook "Analyse des Infiniment Petits" (1696), which included the rule for limits bearing his name (though discovered by Johann Bernoulli). Brook Taylor developed Taylor series (1715) using higher-order derivatives. Colin Maclaurin developed special cases for expansions around zero. Augustin-Louis Cauchy (1823) placed differentiation on a rigorous foundation using limits. The modern notation and rules were refined by Euler, Lagrange, and Laplace. In the 20th century, differentiation was extended to vector-valued functions (Fréchet derivative) and distributions (Schwartz).