derivative notation

This is the list of known standard representations and their nuances.

dudv,dfdx,dydx- The most common notation, this is read as the derivative of u with respect to v. ExponentsPlanetmathPlanetmath relate which derivative, for example, d2ydx2 is the second derivative of y with respect to x.
 f(x),f(𝐱),y′′- This is read as f prime of x. The number of primes tells the derivative, ie. f′′′(x) is the third derivative of f(x) with respect to x. Note that in higher dimensionsPlanetmathPlanetmathPlanetmath, this may be a tensor of a rank equal to the derivative.
 Dxf(𝐱),Fy(𝐱),fxy(𝐱)- These notations are rather arcane, and should not be used generally, as they have other meanings. For example Fy can easily by the y componentMathworldPlanetmathPlanetmathPlanetmath of a vector-valued functionPlanetmathPlanetmath. The subscript in this case means “with respect to”, so Fyy would be the second derivative of F with respect to y.
 D1f(𝐱),F2(𝐱),f12(𝐱)- The subscripts in these cases refer to the derivative with respect to the nth variable. For example, F2(x,y,z) would be the derivative of F with respect to y. They can easily represent higher derivatives, ie. D21f(𝐱) is the derivative with respect to the first variable of the derivative with respect to the second variable.
 uv,fx- The partial derivativeMathworldPlanetmath of u with respect to v. This symbol can be manipulated as in dudv for higher partials.
 ddv,v- This is the operator version of the derivative. Usually you will see it acting on something such as ddv(v2+3u)=2v.
 [𝐉𝐟(𝐱)],[𝐃𝐟(𝐱)]- The first of these represents the of 𝐟, which is a matrix of partial derivatives such that


where fn represents the nth function of a vector valued function. The second of these notations represents the derivative matrix, which in most cases is the Jacobian, but in some cases, does not exist, even though the Jacobian exists. Note that the directional derivativeMathworldPlanetmath in the direction v is simply [𝐉𝐟(𝐱)]v.

Title derivative notation
Canonical name DerivativeNotation
Date of creation 2013-03-22 11:58:27
Last modified on 2013-03-22 11:58:27
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Topic
Classification msc 26-00
Related topic Derivative
Related topic GradientMathworldPlanetmath
Related topic PartialDerivative
Related topic DirectionalDerivative
Related topic JacobianMatrix
Related topic LeibnizNotationForVectorFields