In multivariable calculus, in the context of differential equations defined by a vector valued function Rn to Rm, the Fréchet derivative A is a linear operator on R considered as a vector space over itself, and corresponds to the best linear approximation of a function.
[1] On the exterior algebra of differential forms over a smooth manifold, the exterior derivative is the unique linear map which satisfies a graded version of the Leibniz law and squares to zero.
In R3, the gradient, curl, and divergence are special cases of the exterior derivative.
In differential topology, a vector field may be defined as a derivation on the ring of smooth functions on a manifold, and a tangent vector may be defined as a derivation at a point.
This allows the abstraction of the notion of a directional derivative of a scalar function to general manifolds.
For manifolds that are subsets of Rn, this tangent vector will agree with the directional derivative.
In Riemannian geometry, the existence of a metric chooses a unique preferred torsion-free covariant derivative, known as the Levi-Civita connection.
Weak derivatives are particularly useful in the study of partial differential equations, and within parts of functional analysis.
Higher derivatives can also be defined for functions of several variables, studied in multivariable calculus.
One of the subtle points is that the higher derivatives are not intrinsically defined, and depend on the choice of the coordinates in a complicated fashion (in particular, the Hessian matrix of a function is not a tensor).
Nevertheless, higher derivatives have important applications to analysis of local extrema of a function at its critical points.
For an advanced application of this analysis to topology of manifolds, see Morse theory.
In quaternionic analysis, derivatives can be defined in a similar way to real and complex functions.
A derivation is a linear map on a ring or algebra which satisfies the Leibniz law (the product rule).
For example, the formal derivative of a polynomial over a commutative ring R is defined by The mapping
This information is a tuple that contains a binary indicator of whether the child is on the left or right, the value at the parent, and the sibling subtree.
This concept of a derivative of a type has practical applications, such as the zipper technique used in functional programming languages.
A differential operator combines several derivatives, possibly of different orders, in one algebraic expression.
This is especially useful in considering ordinary linear differential equations with constant coefficients.
is a second order linear constant coefficient differential operator acting on functions of x.
The key idea here is that we consider a particular linear combination of zeroth, first and second order derivatives "all at once".
This allows us to think of the set of solutions of this differential equation as a "generalized antiderivative" of its right hand side 4x − 1, by analogy with ordinary integration, and formally write
Combining derivatives of different variables results in a notion of a partial differential operator.
By means of the Fourier transform, pseudo-differential operators can be defined which allow for fractional calculus.
This is an extension of the directional derivative to an infinite dimensional vector space.
They can be used to define an analogue of exterior derivative from differential geometry that applies to arbitrary algebraic varieties, instead of just smooth manifolds.
In p-adic analysis, the usual definition of derivative is not quite strong enough, and one requires strict differentiability instead.
The H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.
Along with suitably defined analogs to the exponential function, logarithms and others the derivative can be used to develop notions of smoothness, analycity, integration, Taylor series as well as a theory of differential equations.
[4] It may be possible to combine two or more of the above different notions of extension or abstraction of the original derivative.