Derivative

Definition

In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. The derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.

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(ε, δ)-definition of limit Abel's test Absolute value Acceleration Adequality Almost everywhere Alternating series Alternating series test Antiderivative Arc length Arithmetic derivative Arithmetico-geometric sequence Arithmetico–geometric sequence Banach space Bernoulli number Binomial series Binomial theorem Brook Taylor CRC Press Calculus Calculus (Apostol books)Calculus Made Easy Calculus of variations Calculus on Euclidean space Canadian Mathematical Bulletin Cauchy condensation test Cauchy–Riemann equations Chain rule Colin Maclaurin Complex analysis Complex function Complex number Concave function Constant function Constant of integration Continuous function Contour integral Contour integration Convergence tests Covariant derivative Curl (mathematics)Curvature Dependent and independent variables Derivation (algebra)Derivation (differential algebra)Derivative (disambiguation)Derivative work Differentiability class Differentiable Differentiable function Differential (infinitesimal)Differential (mathematics)Differential algebra Differential calculus Differential equation Differential form Differential geometry Differential geometry of curves Differential geometry of surfaces Differential of a function Differential operator Differentiation rules Direct comparison test Directional derivative Dirichlet's test Disc integration Distribution (mathematics)Divergence Divergence theorem Domain of a function Dover Publications E (mathematical constant)Edwin Hewitt Encyclopedia of Mathematics Eric W. Weisstein Euler substitution Euler–Maclaurin formula European Mathematical Society Exponential function Exterior derivative Extreme value theorem Factorial Faà di Bruno's formula Field (mathematics)Field extension Finite difference First derivative test Florian Cajori Fluxion Fourier analysis Fourier series Fractional calculus Frank Wilson Warner Free variables and bound variables Fritz John Fréchet derivative Fubini's theorem Function (mathematics)Function composition Function of a real variable Function of several real variables Functional analysis Functional derivative Functions of several real variables Fundamental Theorem of Line Integrals Fundamental theorem of calculus Gabriel's horn Gateaux derivative General Leibniz rule Generality of algebra Generalizations of the derivative Generalized Stokes' theorem Generalized Stokes theorem Generalized function Geometric calculus Geometric series George B. Thomas Gilbert Strang Glossary of calculus Gottfried Leibniz Gottfried Wilhelm Leibniz Gradient Gradient theorem Gradient vector Graph of a function Green's theorem Greg Gbur Harmonic analysis Harmonic series (mathematics)Heaviside cover-up method Helmholtz decomposition Hessian matrix Higher order derivative History of calculus Holomorphic function Hypercomplex analysis Hyperreal number Ideal (ring theory)Implicit differentiation Improper integral Indeterminate form Infinite series Infinitesimal Infinitesimal calculus Infinity Integer Integral Integral calculus Integral equation Integral of inverse functions Integral of secant cubed Integral of the secant function Integral test for convergence Integral transform Integration Bee Integration by parts Integration by reduction formulae Integration by substitution Integration using Euler's formula Integro-differential equation Interval (mathematics)Inverse function rule Inverse function theorem Inverse trigonometric functions Isaac Newton Jacobian matrix Jacobian matrix and determinant James Stewart (mathematician)Jerk (physics)Joel Hass Joseph-Louis Lagrange Jump discontinuity Khan Academy L'Hôpital's rule Lagrange multiplier Laplace operator Larry Gonick Law of Continuity Lawrence Evans Least-squares spectral analysis Lebesgue integration Leibniz's notation Leibniz integral rule Leibniz notation Leonhard Euler Lie derivative Limit (mathematics)Limit comparison test Limit of a function Limit of a sequence Limit of distributions Line integral Linear approximation Linear function Linear map Linear transformation Linearity of differentiation Lipschitz function List of calculus topics List of integrals of exponential functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of inverse trigonometric functions List of integrals of irrational algebraic functions List of integrals of logarithmic functions List of integrals of rational functions List of integrals of trigonometric functions List of limits Lists of integrals Logarithm Logarithmic derivative Logarithmic differentiation Louis François Antoine Arbogast Maclaurin series Malliavin calculus Manifold MathWorld Mathematical analysis Mathematical notation Mathematics Matrix (mathematics)Matrix calculus Maximum and minimum Meager set Mean value theorem Measure (mathematics)Method of Fluxions Monotone function Multiple integral Multiplicative inverse Multivariable calculus Natural logarithm Newton's method Newton's notation Newton's notation for differentiation Non-standard analysis Nonelementary integral Nonstandard analysis Notation for differentiation One-sided limit Open interval Order of approximation Order of integration (calculus)Ordinary differential equation P-adic analysis P-adic number Parametric curve Partial derivative Partial differential equation Partial fractions in integration Pearson PLC Pearson Prentice Hall Physics Polynomial Power rule Power series Precalculus Prime (symbol)Prime factorization Prime mark Product rule Proof that 22/7 exceeds π Pushforward (differential)Quadratic integral Quaternionic analysis Quotient rule Radian Ratio test Real-valued function Real analysis Real number Regiomontanus' angle maximization problem Related rates Representation theory Reynolds transport theorem Richard Courant Riemann integral Ring (mathematics)Risch algorithm Rolle's theorem Root test Secant line Second derivative Second derivative test Series (mathematics)Shell integration Silvanus P. Thompson Slope Smooth manifold Smooth surface Smoothness Special functions Springer-Verlag Standard part function Stationary point Stefan Banach Steinmetz solid Step function Stirling's approximation Stochastic calculus Stochastic differential equation Stokes' theorem Subscript and superscript Sum rule in differentiation Surface integral Symmetry of second derivatives Table of derivatives Tangent Tangent bundle Tangent half-angle substitution Tangent line Tangent space Taylor's theorem Taylor series Telescoping series Tensor Tensor calculus Term test The Method of Mechanical Theorems Third derivative Time Time scale calculus Tom M. Apostol Total derivative Trapezoidal rule Trigonometric functions Trigonometric substitution Vector-valued function Vector (geometric)Vector calculus Vector calculus identities Vector field Vector space Velocity Vertical tangent Vojtěch Jarník Volume integral Weak derivative Weierstrass function Wolfram Alpha ∂

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