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Mathematical function whose derivative exists
function f is said to be differentiable on a subset U⊆ℝ if it is differentiable at every point in U, and f is said to be continuously differentiable if
Differentiable_function
Function that is continuous everywhere but differentiable nowhere
Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere
Weierstrass function (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
Degree of differentiability of a function or map
value function f(x)=|x| has class C0, because it is continuous, but not differentiable. Generally, the term smooth function refers to a C∞-function, that
Smoothness
Function defined by multiple sub-functions
that the value of the right sub-function is used in this position. For a piecewise-defined function to be differentiable on a given interval in its domain
Piecewise_function
Complex-differentiable (mathematical) function
mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each
Holomorphic_function
Theorem in mathematics
versions of the inverse function theorem for holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces
Inverse_function_theorem
Rules for computing derivatives of functions
and g are n-times differentiable, then: dndxn[f(x)g(x)]=∑k=0n(nk)dn−kdxn−kf(x)dkdxkg(x). Differentiable function – Mathematical function whose derivative
Differentiation_rules
Manifold upon which it is possible to perform calculus
another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold
Differentiable_manifold
Theorem in real analysis
analysis, Rolle's theorem (or lemma) states that a real-valued differentiable function which attains equal values at two distinct points must have a stationary
Rolle's_theorem
Mathematical functions which are smooth but not analytic
In real analysis, a smooth function is infinitely differentiable at each point in its domain, while a real analytic function is, at each point in its domain
Non-analytic_smooth_function
On converting relations to functions of several real variables
Allendoerfer, Carl B. (1974). "Theorems about Differentiable Functions". Calculus of Several Variables and Differentiable Manifolds. New York: Macmillan. pp. 54–88
Implicit_function_theorem
Formula for the derivative of an inverse function
calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the
Inverse_function_rule
Mathematical process of finding the derivative of a trigonometric function
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change
Differentiation of trigonometric functions
Differentiation_of_trigonometric_functions
Mathematical relation consisting of a multi-variable function equal to zero
an implicit function that is differentiable in some small enough neighbourhood of (a, b); in other words, there is a differentiable function f that is defined
Implicit_function
Instantaneous rate of change (mathematics)
Higher-order derivatives are the result of differentiating a function repeatedly. Given that f is a differentiable function, the derivative of f is the first derivative
Derivative
Real function with secant line between points above the graph itself
that interval. If a function is differentiable and convex then it is also continuously differentiable. A differentiable function of one variable is convex
Convex_function
Study of rates of change
approximation to a differentiable function near a point. In this sense, differentiation is closely related to the differential. For functions of several variables
Differential_calculus
Derivative defined on normed spaces
function that is Fréchet differentiable at a point is necessarily continuous there and sums and scalar multiples of Fréchet differentiable functions are
Fréchet_derivative
Measure of local oscillation behavior
C_{c}^{1}(\Omega ,\mathbb {R} ^{n})} is the set of continuously differentiable vector functions of compact support contained in Ω {\displaystyle \Omega }
Total_variation
Differentiable function whose derivative is not Riemann integrable
properties: V is differentiable everywhere The derivative V ′ is bounded everywhere The derivative is not Riemann-integrable. The function is defined by
Volterra's_function
Property of a mathematical function
zero (note that this indicator function is not left differentiable at zero). If a real-valued, differentiable function f, defined on an interval I of
Semi-differentiability
Strong form of uniform continuity
to 1. Lipschitz continuous functions that are everywhere differentiable but not continuously differentiable The function f ( x ) = { x 2 sin ( 1 /
Lipschitz_continuity
Point where the derivative of a function is zero or undefined (in certain cases)
y-axis. The function f(x)=x2/3 is defined for all x and differentiable for x ≠ 0, with the derivative f′(x)=2x−1/33. Since f is not differentiable at x = 0
Critical_point_(mathematics)
Topics referred to by the same term
Weierstrass function may refer to: Weierstrass function (nowhere-differentiable function) Weierstrass sigma, zeta, and eta functions Weierstrass elliptic
Weierstrass_function
Multivariate derivative (mathematics)
gradient ∇f of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) whose value at a point p gives
Gradient
Branch of mathematics studying functions of a complex variable
holomorphic functions, which are differentiable functions of a complex variable. In contrast with a differentiable real function, a holomorphic function is always
Complex_analysis
Matrix of partial derivatives of a vector-valued function
be differentiable for its Jacobian matrix to be defined, since only its first-order partial derivatives are required to exist. If f is differentiable at
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Counterintuitive mathematical object
Weierstrass function, a function that is continuous everywhere but differentiable nowhere. The sum of a differentiable function and the Weierstrass function is
Pathological_(mathematics)
Analyzes the topology of a manifold by studying differentiable functions on that manifold
by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold
Morse_theory
Theorem in mathematics
theorem) is a theorem about differentiable functions, roughly stating that the average rate of change of such a function over an interval is equal to
Mean_value_theorem
Approximation of a function by a polynomial
k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial. For a smooth function, the Taylor
Taylor's_theorem
Mathematical operation in calculus
can be found by similar methods. Let F {\displaystyle F} be a differentiable function of two variables, and suppose that an equation F ( x , y ) = F
Implicit_differentiation
Formula in calculus
formula that expresses the derivative of the composition of two differentiable functions z and y in terms of the derivatives of z and y. More precisely
Chain_rule
Type of function in mathematics
complex differentiable at every point of the set. For this reason, in complex analysis the terms analytic function and holomorphic function are often
Analytic_function
Operation in differential calculus
differentiable at x = 0, but is symmetrically differentiable there with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does
Symmetric_derivative
Notion in calculus
and differentiable functions f and g, d ( a f + b g ) = a d f + b d g . {\displaystyle d(af+bg)=a\,df+b\,dg.} Product rule: For two differentiable functions
Differential_of_a_function
Generalized function whose value is zero everywhere except at zero
delta function is to a differentiable manifold where most of its properties as a distribution can also be exploited because of the differentiable structure
Dirac_delta_function
Type of derivative in mathematics
approximation to f(a+h)−f(a). The function f is differentiable if and only if each of its components fi:U→ℝ is differentiable, so when studying derivatives
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Method of differentiating single-term polynomials
differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the
Power_rule
Association of one output to each input
century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized
Function_(mathematics)
Zero of the derivative of a function
a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally
Stationary_point
Evaluates a line integral through a gradient field using the original scalar field
rather than just the real line. If φ : U ⊆ Rn → R is a differentiable function and γ a differentiable curve in U which starts at a point p and ends at a point
Gradient_theorem
Theorem in convex analysis
derivative of the maximum of a (not necessarily convex) directionally differentiable function. An extension to more general conditions was proven 1971 by Dimitri
Danskin's_theorem
About maxima and minima of functions
boundary of A non-differentiable: f is not differentiable at x0 stationary point: f′(x0)=0 The function f(x)=x3 has no extrema. The function f(x)=x3−x has
Interior_extremum_theorem
Negative of a convex function
(y,f(y)). A function f is quasiconcave if the upper contour sets of the function S(a)={x:f(x)≥a} are convex sets. A differentiable function f is (strictly)
Concave_function
Functions in mathematics
the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle f\colon U\to \mathbb
Harmonic_function
Mathematical function with no sudden changes
everywhere continuous but nowhere differentiable. The derivative f ′ ( x ) {\displaystyle f'(x)} of a differentiable function f ( x ) {\displaystyle f(x)}
Continuous_function
Description of continuous random distribution
density function if its cumulative distribution function F(x) is absolutely continuous. In this case: F is almost everywhere differentiable, and its
Probability_density_function
Generalization of the concept of directional derivative
redirect targets Differentiable vector-valued functions from Euclidean space – Differentiable function in functional analysis Differentiation in Fréchet spaces –
Gateaux_derivative
Method in statistics
when the random variable being considered can be defined as a differentiable function of a random variable which is asymptotically Gaussian. More generally
Delta_method
Polynomial function of degree 3
values of a function and its derivative at some sampling points, one can interpolate the function with a continuously differentiable function, which is
Cubic_function
Differentiable function whose derivative is everywhere injective
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly
Immersion_(mathematics)
Mathematical analysis of real numbers
degrees of regularity. A function may be continuous but nowhere differentiable, differentiable but not continuously differentiable, or smooth (having derivatives
Real_analysis
Concept in mathematical analysis
derivative is a real-valued function of one real variable that is the derivative of an everywhere differentiable function and that vanishes in a dense
Pompeiu_derivative
exist functions which are not in any Baire class. Examples: The derivative of any differentiable function is of class 1. An example of a differentiable function
Baire_function
Indefinite integral
function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function
Antiderivative
Instantaneous rate of change of the function
chain rule: If g is differentiable at p and h is differentiable at g(p), then ∇𝐯(h∘g)(𝐩)=h′(g(𝐩))∇𝐯g(𝐩). Let M be a differentiable manifold and p a
Directional_derivative
Mathematical idealization of the trace left by a moving point
regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve. A plane
Curve
Concept in mathematics
iterative optimization algorithm for finding a local minimum of a differentiable function. It generalizes algorithms such as gradient descent and multiplicative
Mirror_descent
Mathematical theorem
fact smooth functions are another valid domain. In terms of the total derivative, the second derivative of a twice-differentiable function f : X → Y {\displaystyle
Symmetry of second derivatives
Symmetry_of_second_derivatives
Formula for the derivative of a ratio of functions
derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)g(x), where both f and g are differentiable and g(x)≠0.
Quotient_rule
Mapping which preserves all topological properties of a given space
θ)↦(ρmax(|cosθ|,|sinθ|),θ). The graph of a differentiable function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is a
Homeomorphism
Type of infinitesimal in calculus
for some differentiable function Q {\displaystyle Q} in an orthogonal coordinate system (hence Q {\displaystyle Q} is a multivariable function whose variables
Exact_differential
Method of mathematical differentiation
implemented, at least in part, in the differentiation of almost all differentiable functions, providing that these functions are non-zero. The method is used
Logarithmic_differentiation
Method for finding stationary points of a function
Newton–Raphson) is an iterative method for finding the roots of a differentiable function f {\displaystyle f} , which are solutions to the equation f ( x
Newton's method in optimization
Newton's_method_in_optimization
Technique in integral evaluation
requirement that φ be continuously differentiable can be replaced by the weaker assumption that φ be merely differentiable and have a continuous inverse.
Integration_by_substitution
Mathematical transformation
of differentiable manifolds). This definition is equivalent to the modern mathematicians' definition as long as f {\displaystyle f} is differentiable and
Legendre_transformation
Algorithm for finding zeros of functions
Suppose that the function f has a zero at α, i.e., f(α) = 0, and f is differentiable in a neighborhood of α. If f is continuously differentiable and its derivative
Newton's_method
Characteristic property of holomorphic functions
Conversely, if the functions u and v are (real) differentiable at z and satisfy the Cauchy-Riemann equations there, then f is complex-differentiable at z. In this
Cauchy–Riemann_equations
Continuously differentiable function: differentiable, with continuous derivative. Smooth function: Has derivatives of all orders. Lipschitz function, Holder
List_of_types_of_functions
Calculus of vector-valued functions
are used to replace complicated functions with linear functions that are almost the same. Given a differentiable function f(x, y) with real values, one
Vector_calculus
Numerical calculations carrying along derivatives
to evaluate the partial derivative of a function specified by a computer program. Automatic differentiation enables the simultaneous computation of the
Automatic_differentiation
Concept in complex analysis
variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit
Wirtinger_derivatives
Order-preserving mathematical function
{\displaystyle f} is a monotonic function defined on an interval I {\displaystyle I} , then f {\displaystyle f} is differentiable almost everywhere on I {\displaystyle
Monotonic_function
Equivalence class of objects sharing local properties at a point in a topological space
local properties, such as continuity, differentiability etc., so it makes sense to talk about a differentiable or analytic germ, etc. Similarly for subsets:
Germ_(mathematics)
In vector calculus, an invex function is a differentiable function f:ℝn→ℝ for which there exists a vector-valued function η such that f(x)−f(u)≥η(x,u)⋅∇f(u)
Invex_function
Differentiable function in functional analysis
discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector
Differentiable vector-valued functions from Euclidean space
Differentiable_vector-valued_functions_from_Euclidean_space
Formula for the derivative of a product
is differentiable but only says what its derivative is if it is differentiable.) Let h(x) = f(x)g(x) and suppose that f and g are each differentiable at
Product_rule
Theorem on holomorphic functions
difference between holomorphy and real-differentiability. On the real line, for example, the differentiable function f ( x ) = x 2 {\displaystyle f(x)=x^{2}}
Open mapping theorem (complex analysis)
Open_mapping_theorem_(complex_analysis)
Mathematical model of the time dependence of a point in space
equations. If Φ is continuously differentiable the system is called a differentiable dynamical system. The function f is therefore a "smooth" mapping
Dynamical_system
Computation of an antiderivatives
integral, of a given function f(x), i.e. to find a formula for a differentiable function F(x) such that dFdx=f(x). The family of all functions that satisfy this
Symbolic_integration
Mathematical method in calculus
v} to be continuously differentiable. Integration by parts works if u {\displaystyle u} is absolutely continuous and the function designated v ′ {\displaystyle
Integration_by_parts
Objects that generalize functions
analysis, distributions are a type of generalized function that makes it possible to differentiate functions whose derivatives do not exist in the classical
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Type of function
its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has
Pseudoconvex_function
Mathematical function, denoted exp(x) or e^x
definitions of the exponential function, although of very different nature. The exponential function is the unique differentiable function that equals its derivative
Exponential_function
Mathematical definition of point elasticity
mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output) at point
Elasticity_of_a_function
Point where the curvature of a curve changes sign
the curvature changes its sign. For example, the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative
Inflection_point
Artificial neural network node function
Continuously differentiable This property is desirable for enabling gradient-based optimization methods (ReLU is not continuously differentiable and has some
Activation_function
Isomorphism of differentiable manifolds
continuously differentiable. Given two differentiable manifolds M {\displaystyle M} and N {\displaystyle N} , a continuously differentiable map f : M →
Diffeomorphism
Topological space that locally resembles Euclidean space
additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric
Manifold
Mathematical idealization of the surface of a body
surface). A differentiable surface is a surfaces that is a differentiable manifold (see § Differentiable surface). Every differentiable surface is a
Surface_(mathematics)
Form of projection
\mathbb {R} ,\ i=1,\dots ,n} are possibly non-differentiable convex functions. The lack of differentiability rules out conventional smooth optimization techniques
Proximal_gradient_method
Largest and smallest value taken by a function at a given point
points on the boundary, and take the greatest (or least) one. For differentiable functions, Fermat's theorem states that local extrema in the interior of
Maximum_and_minimum
Topics referred to by the same term
prime-counting function π(x); see Prime-counting function § Exact form. Almost nowhere differentiable Riemann function, which the Weierstrass function (nowhere-differentiable
Riemann_function
Programming paradigm
Differentiable programming is a programming paradigm in which a numeric computer program can be differentiated throughout via automatic differentiation
Differentiable_programming
Function with a multiplicative scaling behaviour
every maximal continuously differentiable solution of this partial differentiable equation is a positively homogeneous function of degree k, defined on a
Homogeneous_function
requiring x ≠ y {\displaystyle x\neq y} . A strictly differentiable function is obviously differentiable, but the converse is wrong, as can be seen from the
Strict_differentiability
inverse function see also List of differential geometry topics Differentiable manifold Differentiable structure Submersion – a differentiable map between
List_of_real_analysis_topics
Optimization algorithm
It is a first-order iterative algorithm for minimizing a differentiable multivariate function. The idea is to take repeated steps in the opposite direction
Gradient_descent
All derivatives have the intermediate value property
interval. Let I be an open interval, and let f:I→ℝ be a real-valued differentiable function. Then f′ has the intermediate value property: If a and b are points
Darboux's_theorem_(analysis)
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