directional derivative

Let U be an open set in n and f:U is a differentiable function. If uU and vn, then the directional derivativeMathworldPlanetmathPlanetmath of f in the direction of v is


In other words, (Dvf)(u) measures how f changes in the direction of v from u.


(Dvf)(u) = limh0f(u+hv)-f(u)h
= Df(u)v,

where Df is the Jacobian matrix of f.


Let uU.

  1. 1.

    Dvf is linear in v. If v,wn and λ,μ, then


    In particular, D0f=0.

  2. 2.

    If f is twice differentiableMathworldPlanetmath and v,wn, then

    DvDwf(u) = 2stf(u+sv+tw)|s=0,
    = vTHessf(u)w,

    where Hess is the Hessian matrix of f.


For example, if f(xyz)=x2+3y2z, and we wanted to find the derivative at the point 𝐚=(123) in the direction v=[111], our equation would be

limh01h((1+h)2+3(2+h)2(3+h)-37) = limh01h(3h3+37h2+50h)
= limh03h2+37h+50=50
Title directional derivative
Canonical name DirectionalDerivative
Date of creation 2013-03-22 11:58:37
Last modified on 2013-03-22 11:58:37
Owner matte (1858)
Last modified by matte (1858)
Numerical id 15
Author matte (1858)
Entry type Definition
Classification msc 26B12
Classification msc 26B10
Synonym derivative with respect to a vector
Synonym partial derivativeMathworldPlanetmath with respect to a vector
Related topic PartialDerivative
Related topic Derivative
Related topic DerivativeNotation
Related topic JacobianMatrix
Related topic GradientMathworldPlanetmath
Related topic FixedPointsOfNormalFunctions
Related topic HessianMatrix