**mathematics**- In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry. (wikipedia.org)
- In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. (wikipedia.org)
- In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. (wikipedia.org)
- In mathematics, the gradient is a multi-variable generalization of the derivative. (wikipedia.org)
- In mathematics, in the area of combinatorics, the q-derivative, or Jackson derivative, is a q-analog of the ordinary derivative, introduced by Frank Hilton Jackson. (wikipedia.org)
- In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. (wikipedia.org)
- In mathematics, the Hasse derivative is a generalisation of the derivative which allows the formulation of Taylor's theorem in coordinate rings of algebraic varieties. (wikipedia.org)
- In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. (wikipedia.org)
- In mathematics, the reduced derivative is a generalization of the notion of derivative that is well-suited to the study of functions of bounded variation. (wikipedia.org)

**calculus**- An important case is the variational derivative in the calculus of variations. (wikipedia.org)
- Matrix calculus refers to a number of different notations that use matrices and vectors to collect the derivative of each component of the dependent variable with respect to each component of the independent variable. (wikipedia.org)
- For this reason, a generalisation of the lemma can be used in the definition of differentiability in multivariable calculus. (wikipedia.org)
- A similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus. (wikipedia.org)
- It can be viewed as a generalization of the total derivative of ordinary calculus. (wikipedia.org)
- Derivative (generalizations) Jackson integral Q-exponential Q-difference polynomials Quantum calculus Tsallis entropy F. H. Jackson (1908), On q-functions and a certain difference operator, Trans. (wikipedia.org)
- It allows a generalization of the single-variable fundamental theorem of calculus to higher dimensions, in a different vein than the generalization that is Stokes' theorem. (wikipedia.org)
- The reason that one might call the Darboux derivative a more natural generalization of the derivative of single-variable calculus is this. (wikipedia.org)
- In single-variable calculus, the derivative f ′ {\displaystyle f'} of a function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } assigns to each point in the domain a single number. (wikipedia.org)

**lemma**- The lemma asserts that the existence of this derivative implies the existence of a function φ {\displaystyle \varphi } such that lim h → 0 φ ( h ) = 0 and f ( a + h ) = f ( a ) + f ′ ( a ) h + φ ( h ) h {\displaystyle \lim _{h\to 0}\varphi (h)=0\qquad {\text{and}}\qquad f(a+h)=f(a)+f'(a)h+\varphi (h)h} for sufficiently small but non-zero h. (wikipedia.org)
- Bihari I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. (degruyter.com)

**convective derivative**- The convective derivative takes into account changes due to time dependence and motion through space along vector field. (wikipedia.org)
- Generally the convective derivative of the field u·∇y, the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field u·(∇y), or as involving the streamline directional derivative of the field (u·∇) y, leading to the same result. (wikipedia.org)
- Confusingly, sometimes the name "convective derivative" is used for the whole material derivative D/Dt, instead for only the spatial term u·∇, which is also a redundant nomenclature. (wikipedia.org)
- In the nonredundant nomenclature the material derivative only equals the convective derivative for absent flows. (wikipedia.org)

**derivatives**- The second derivative implied by a parametric equation is given by by making use of the quotient rule for derivatives. (wikipedia.org)
- In this case, M is the unique derivative (or total derivative, to distinguish from the directional and partial derivatives) of f at a. (wikipedia.org)
- According to the more general manifold ideas of derivatives, the derivative assigns to each point in the domain a linear map from the tangent space at the domain point to the tangent space at the image point. (wikipedia.org)
- The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist. (wikipedia.org)
- Although functions of bounded variation have derivatives in the sense of Radon measures, it is desirable to have a derivative that takes values in the same space as the functions themselves. (wikipedia.org)
- Now, to generalize this to fractional order derivatives, we just have to generalize the coefficients, which must be similar to the generalization of the expansion of ##(x-1)^n## to fractional exponents. (physicsforums.com)

**differentiable at x0**- More generally, if x0 is a point in the domain of a function f, then f is said to be differentiable at x0 if the derivative f ′(x0) exists. (wikipedia.org)
- If f is Fréchet differentiable at x0, then by the chain rule, f is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at x0. (wikipedia.org)

**displaystyle**- Since the line L ( z ) = f ′ ( x ) z − f ′ ( x ) x + f ( x ) {\displaystyle L(z)=f'(x)z-f'(x)x+f(x)} is tangent to the original function at the point ( x , f ( x ) ) , {\displaystyle (x,f(x)),} the derivative can be seen as a way to find the best linear approximation of a function. (wikipedia.org)
- The first derivative implied by these parametric equations is d y d x = d y / d t d x / d t = y ˙ ( t ) x ˙ ( t ) , {\displaystyle {\frac {dy}{dx}}={\frac {dy/dt}{dx/dt}}={\frac {{\dot {y}}(t)}{{\dot {x}}(t)}},} where the notation x ˙ ( t ) {\displaystyle {\dot {x}}(t)} denotes the derivative of x with respect to t, for example. (wikipedia.org)
- Substituting these into the formula for the parametric derivative, we obtain d y d x = y ˙ x ˙ = 3 8 t , {\displaystyle {\frac {dy}{dx}}={\frac {\dot {y}}{\dot {x}}}={\frac {3}{8t}},} where x ˙ {\displaystyle {\dot {x}}} and y ˙ {\displaystyle {\dot {y}}} are understood to be functions of t. (wikipedia.org)
- The lower Dini derivative, f′−, is defined by f − ′ ( t ) ≜ lim inf h → 0 + f ( t + h ) − f ( t ) h , {\displaystyle f'_{-}(t)\triangleq \liminf _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}},} where lim inf is the infimum limit. (wikipedia.org)
- This type of generalized derivative can be seen as the derivative of a scalar, f, with respect to a vector, x {\displaystyle \mathbf {x} } and its result can be easily collected in vector form. (wikipedia.org)
- The gradient of f is defined as the unique vector field whose dot product with any unit vector v at each point x is the directional derivative of f along v. That is, ( ∇ f ( x ) ) ⋅ v = D v f ( x ) . {\displaystyle {\big (}\nabla f(x){\big )}\cdot \mathbf {v} =D_{\mathbf {v} }f(x). (wikipedia.org)
- Formally, in terms of Lagrange's shift operator in logarithmic variables, it amounts to the operator D q = 1 x q d d ( ln x ) − 1 q − 1 , {\displaystyle D_{q}={\frac {1}{x}}~{\frac {q^{d~~~ \over d(\ln x)}-1}{q-1}}~,} which goes to the plain derivative, → d⁄dx, as q → 1. (wikipedia.org)
- Note that lim q → 1 [ n ] q = n {\displaystyle \lim _{q\to 1}[n]_{q}=n} so the ordinary derivative is regained in this limit. (wikipedia.org)
- One way to justify this convention of retaining only the linear map aspect of the derivative is to appeal to the (very simple) Lie group structure of R {\displaystyle \mathbb {R} } under addition. (wikipedia.org)
- It is apparent that this derivative is dependent on the vector x ˙ ≡ d x d t , {\displaystyle {\dot {\mathbf {x} }}\equiv {\frac {\mathrm {d} \mathbf {x} }{\mathrm {d} t}},} which describes a chosen path x(t) in space. (wikipedia.org)
- For example, if x ˙ = 0 {\displaystyle {\dot {\mathbf {x} }}=\mathbf {0} } is chosen, the time derivative becomes equal to the partial time derivative, which agrees with the definition of a partial derivative: a derivative taken with respect to some variable (time in this case) holding other variables constant (space in this case). (wikipedia.org)
- This makes sense because if x ˙ = 0 {\displaystyle {\dot {\mathbf {x} }}=0} , then the derivative is taken at some constant position. (wikipedia.org)
- The r-th Hasse derivative of Xn is D ( r ) X n = ( n r ) X n − r , {\displaystyle D^{(r)}X^{n}={\binom {n}{r}}X^{n-r},} if n ≥ r and zero otherwise. (wikipedia.org)
- The Hasse derivative is a generalized derivation on k[X] and extends to a generalized derivation on the function field k(X), satisfying an analogue of the product rule D ( r ) ( f g ) = ∑ i = 0 r D ( i ) ( f ) D ( r − i ) ( g ) {\displaystyle D^{(r)}(fg)=\sum _{i=0}^{r}D^{(i)}(f)D^{(r-i)}(g)} and an analogue of the chain rule. (wikipedia.org)
- In Leibniz's notation, an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written d y d x {\displaystyle {\frac {dy}{dx}}} suggesting the ratio of two infinitesimal quantities. (wikipedia.org)
- Again, for this function the symmetric derivative exists at x = 0 {\displaystyle x=0} , while its ordinary derivative does not exist at x = 0 {\displaystyle x=0} , due to discontinuity in the curve there. (wikipedia.org)

**Jacobian**- The Jacobian is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. (wikipedia.org)

**Lagrange's**- In Lagrange's notation, the derivative with respect to x of a function f(x) is denoted f'(x) (read as "f prime of x") or fx′(x) (read as "f prime x of x"), in case of ambiguity of the variable implied by the derivation. (wikipedia.org)

**theorem**- Denjoy-Young-Saks theorem Derivative (generalizations) Khalil, Hassan K. (2002). (wikipedia.org)
- Nevertheless, Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem. (wikipedia.org)
- The symmetric derivative does not obey the usual mean value theorem (of Lagrange). (wikipedia.org)
- A theorem somewhat analogous to Rolle's theorem but for the symmetric derivative was established by in 1967 C.E. Aull, who named it Quasi-Rolle theorem. (wikipedia.org)
- These integrals are shown to be generalizations of the Fourier integral theorem. (ams.org)

**gradient**- For real valued functions from Rn to R (scalar fields), the total derivative can be interpreted as a vector field called the gradient. (wikipedia.org)
- While a derivative can be defined on functions of a single variable, for functions of several variables, the gradient takes its place. (wikipedia.org)
- The gradient is a vector-valued function, as opposed to a derivative, which is scalar-valued. (wikipedia.org)
- Like the derivative, the gradient represents the slope of the tangent of the graph of the function. (wikipedia.org)
- More precisely, when H is differentiable, the dot product of the gradient of H with a given unit vector is equal to the directional derivative of H in the direction of that unit vector. (wikipedia.org)

**partial**- Each entry of this matrix represents a partial derivative, specifying the rate of change of one range coordinate with respect to a change in a domain coordinate. (wikipedia.org)
- In patients misery from epilepsy, the Lyrical preparation is habituated to as a means of additional therapy after feeling an attraction (fond of) seizures, including partial seizures, which are accompanied aside inferior generalization. (dclans.ru)
- In patients misery from epilepsy, the Lyrical preparation is occupied as a means of additional cure for feeling an attraction (partial) seizures, including incomplete seizures, which are accompanied by derivative generalization. (dclans.ru)

**parametric**- For vector-valued functions from R to Rn (i.e., parametric curves), one can take the derivative of each component separately. (wikipedia.org)
- Derivative (generalizations) Derivative for parametric form at PlanetMath.org. (wikipedia.org)

**Banach**- However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions. (wikipedia.org)
- A further generalization for a function between Banach spaces is the Fréchet derivative. (wikipedia.org)

**vectors**- Hence it can be used to push tangent vectors on M forward to tangent vectors on N. The differential of a map φ is also called, by various authors, the derivative or total derivative of φ, and is sometimes itself called the pushforward. (wikipedia.org)

**differentiation**- The process of finding a derivative is called differentiation. (wikipedia.org)
- Differentiation is the action of computing a derivative. (wikipedia.org)

**scalar**- The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector. (wikipedia.org)

**ordinary**- provided that the ordinary n-th derivative of f exists at x = 0. (wikipedia.org)

**functions**- One of the simplest settings for generalizations is to vector valued functions of several variables (most often the domain forms a vector space as well). (wikipedia.org)
- This motivates the following generalization to functions mapping Rm to Rn: ƒ is differentiable at x if there exists a linear operator A(x) (depending on x) such that lim ‖ h ‖ → 0 ‖ f ( x + h ) − f ( x ) − A ( x ) h ‖ ‖ h ‖ = 0. (wikipedia.org)
- The subderivative and subgradient are generalizations of the derivative to convex functions. (wikipedia.org)
- As another example, if we have an n-vector of dependent variables, or functions, of m independent variables we might consider the derivative of the dependent vector with respect to the independent vector. (wikipedia.org)
- Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known. (wikipedia.org)
- For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient. (wikipedia.org)
- thus multivariate generalization functions. (coursera.org)

**tangent**- 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. (wikipedia.org)

**particle**- This is useful, for example, if the vector-valued function is the position vector of a particle through time, then the derivative is the velocity vector of the particle through time. (wikipedia.org)

**notation**- f(a) then there exist a point z in (a, b) where the symmetric derivative is non-negative, or with the notation used above, fs(z) ≥ 0. (wikipedia.org)

**slope**- If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. (wikipedia.org)

**weaker**- The quasi-derivative is a slightly stronger version of the Gâteaux derivative, though weaker than the Fréchet derivative. (wikipedia.org)

**integral**- The rough, heuristic idea is that the exterior derivative is the integral over an infinitesimal parallelepiped. (physicsforums.com)

**equation**- A. Atangana and R. T. Alqahtani , Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Advances in Difference Equations , 2016 (2016), 1-13. (aimsciences.org)

**velocity**- In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field variations of that physical quantity. (wikipedia.org)
- For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. (wikipedia.org)

**precise definition**- The most common approach to turn this intuitive idea into a precise definition is to define the derivative as a limit of difference quotients of real numbers. (wikipedia.org)

**dependent variable**- 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. (wikipedia.org)

**continuous function**- A function f is said to be continuously differentiable if the derivative f'(x) exists and is itself a continuous function. (wikipedia.org)

**2016**- O. J. J. Algahtani , Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model, Chaos Solitons and Fractals , 89 (2016), 552-559. (aimsciences.org)
- B. S. T. Alkahtani , Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos, Solitons and Fractals , 89 (2016), 547-551. (aimsciences.org)
- B. S. T. Alkahtani and A. Atangana , Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order, Chaos Soliton and Fractals , 89 (2016), 539-546. (aimsciences.org)
- A. Atangana and B. S. T. Alkahtani , New model of groundwater owing within a confine aquifer: Application of Caputo-Fabrizio derivative, Arabian Journal of Geosciences , 9 (2016), 3647-3654. (aimsciences.org)

**notations**- Two distinct notations are commonly used for the derivative, one deriving from Leibniz and the other from Joseph Louis Lagrange. (wikipedia.org)

**commonly**- No. 3,393,197 issued to Pachter and Matossian on July 16, 1968 disclose N-substituted-14-hydroxydihydronormorphines, including the N-cyclobutylmethyl derivative, commonly called nalbuphine. (google.com)

**exists**- A function is said symmetrically differentiable at a point x if its symmetric derivative exists at that point. (wikipedia.org)
- i.e. the symmetric derivative exists for rational numbers but not for irrational numbers. (wikipedia.org)

**definition**- Continuity of u need not be assumed, but it follows instead from the definition of the quasi-derivative. (wikipedia.org)

**point**- This derivative encapsulates two pieces of data: the image of the domain point and the linear map. (wikipedia.org)

**temperature**- In which case, the material derivative then describes the temperature change of a certain fluid parcel with time, as it flows along its pathline (trajectory). (wikipedia.org)

**possible**- The result could be collected in an m×n matrix consisting of all of the possible derivative combinations. (wikipedia.org)
- Though the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity. (wikipedia.org)

**linear**- 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. (wikipedia.org)

**equal**- Finally, if f is quasi-differentiable, then it is Gâteaux differentiable and its Gâteaux derivative is equal to its quasi-derivative. (wikipedia.org)

**second**- A function is of class C2 if the first and second derivative of the function both exist and are continuous. (wikipedia.org)

**position**- This static position derivative is called the Eulerian derivative. (wikipedia.org)

**single-variable**- In a certain sense, it is arguably a more natural generalization of the single-variable derivative. (wikipedia.org)

**usual**- If f is differentiable at t, then the Dini derivative at t is the usual derivative at t. (wikipedia.org)

**another**- The resulting derivative is another vector valued function. (wikipedia.org)

**case**- In this case the derivative is represented by a 1-by-1 matrix consisting of the sole entry f'(x). (wikipedia.org)

**value**- Its value is then the derivative ƒ'(x). (wikipedia.org)
- The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). (wikipedia.org)
- The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. (wikipedia.org)