Part of a series of articles about
Calculus
${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds
Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution (trigonometric, tangent half-angle, Euler) Euler's formula Partial fractions Changing order Reduction formulae Differentiating under the integral sign Risch algorithm
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence generalized Stokes Helmholtz decomposition
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian
Advanced Calculus on Euclidean space Generalized functions Limit of distributions
Specialized Fractional Malliavin Stochastic Variations
Miscellaneous Precalculus History Glossary List of topics Integration Bee Mathematical analysis Nonstandard analysis
v t e

The derivative is a fundamental tool of calculus that quantifies the sensitivity of 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. For this reason, 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.^[1] The process of finding a derivative is called differentiation.

There are multiple different notations for differentiation, two of the most commonly used being Leibniz notation and prime notation. Leibniz notation, named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas prime notation is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.

Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

Definition

As a limit

A function of a real variable ${\displaystyle f(x)}$ is differentiable at a point ${\displaystyle a}$ of its domain, if its domain contains an open interval containing ${\displaystyle a}$, and the limit

exists.^[2] This means that, for every positive real number ${\displaystyle \varepsilon }$, there exists a positive real number ${\displaystyle \delta }$ such that, for every ${\displaystyle h}$ such that ${\displaystyle |h|<\delta }$ and ${\displaystyle h\neq 0}$ then ${\displaystyle f(a+h)}$ is defined, and where the vertical bars denote the absolute value. This is an example of the (ε, δ)-definition of limit.^[3]

If the function ${\displaystyle f}$ is differentiable at ${\displaystyle a}$, that is if the limit ${\displaystyle L}$ exists, then this limit is called the derivative of ${\displaystyle f}$ at ${\displaystyle a}$. Multiple notations for the derivative exist.^[4] The derivative of ${\displaystyle f}$ at ${\displaystyle a}$ can be denoted ${\displaystyle f'(a)}$, read as "${\displaystyle f}$ prime of ${\displaystyle a}$"; or it can be denoted ${\textstyle {\frac {df}{dx}}(a)}$, read as "the derivative of ${\displaystyle f}$ with respect to ${\displaystyle x}$ at ${\displaystyle a}$" or "${\displaystyle df}$ by (or over) ${\displaystyle dx}$ at ${\displaystyle a}$". See § Notation below. If ${\displaystyle f}$ is a function that has a derivative at every point in its domain, then a function can be defined by mapping every point ${\displaystyle x}$ to the value of the derivative of ${\displaystyle f}$ at ${\displaystyle x}$. This function is written ${\displaystyle f'}$ and is called the derivative function or the derivative of ${\displaystyle f}$. The function ${\displaystyle f}$ sometimes has a derivative at most, but not all, points of its domain. The function whose value at ${\displaystyle a}$ equals $$Zdroj:https://en.wikipedia.org?pojem=Derivative_(mathematics)
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---