Generalizations of the derivative

thumb|300px|The gradient, represented by the blue arrows, denotes the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high).

notion in calculus

the matrix of all first-order partial derivatives of a vector-valued function

a continuous functional on a space of test functions (Schwartz space), which generalizes the concept of locally integrable functions

instantaneous rate of change of the function

time rate of change of a physical quantity in a moving medium

fundamental result in measure theory that expresses the relationship between two measures defined on the same measurable space

weak derivation

derivative of a tensor field along the flow defined by a vector field

a derivative defined on normed spaces

in differential geometry, a differential operation defined in differential forms that increases the form degree by 1

right|thumb|A convex function (blue) and "subtangent lines" at x_0 (red). In mathematics, the subderivative (or subgradient) generalizes the derivative to convex functions which are not necessarily differentiable. The set of subderivatives at a point is called the subdifferential at that point. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization.

linear approximation of smooth maps on tangent spaces

generalization of the derivative

generalization of the concept of directional derivative, defined for functions between locally convex topological vector spaces

In mathematics, in the area of combinatorics and quantum calculus, the '''q-derivative, or Jackson derivative', is a q''-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's q-integration. For other forms of q-derivative, see .

fundamental construction of differential calculus

function defined on integers in number theory

class of generalisations of the derivative

infinitesimal version of a Lie groupoid: manifold M with vector bundle E, vector bundle map ρ: E→TM, and a Lie bracket on sections of E, so that [s,ft] = (ρ(s)f)t+f[s,t] for any function f: M→ℝ and sections s, t of E

In fractional calculus, an area of mathematical analysis, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the q-differintegral of f, here denoted by \mathbb{D}^q f is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the q-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several definitions of the differintegral.