Math 2.0

Real functions Proposition

Derivative gives the directional derivatives

Builds on Differentiable function of several variablesHaving a directional derivativePartial derivative

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, let f ⁣:U→ℝmf \colon U \to \mathbb{R}^m, and let a∈Ua \in U. Suppose that ff is differentiable at aa with derivative Df(a)Df(a). Then for each v∈ℝnv \in \mathbb{R}^n, ff has the directional derivative Df(a)(v)Df(a)(v) at aa in the direction vv: lim⁡t→0f(a+tv)−f(a)t=Df(a)(v).\lim_{t \to 0} \frac{f(a + t v) - f(a)}{t} = Df(a)(v).

Remarks

For v=eiv = e_i this gives ∂f∂xi(a)=Df(a)(ei)\dfrac{\partial f}{\partial x_i}(a) = Df(a)(e_i). So the partial derivatives of the components of ff are the entries of the matrix of Df(a)Df(a). For a real-valued ff, it gives Dvf(a)=∇f(a)⋅vD_v f(a) = \nabla f(a) \cdot v, where ∇f(a)\nabla f(a) is the gradient. In particular, Dvf(a)D_v f(a) is linear in vv. A function whose directional derivatives are not linear in vv is not differentiable.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHaving a directional derivative, definition, no evidence levelHaving a directional …definitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionDerivative gives the directional derivatives, statement, no evidence levelDerivative gives the …statement
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  1. Euclidean space object
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  8. Derivative gives the directional derivatives statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Directional_derivative, For differentiable functions
  2. Web https://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative, Finite dimensions
About this page
Tag0A6PMR20N4
URIm2://0A6PMR20N4
Kindstatement, proposition
WritingW0 stub
MSC26B05
StatusThis copy of the site has no build data. Run m2 status 0A6PMR20N4.
UsesDifferentiable function of several variables, Having a directional derivative, Partial derivative

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