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{{refimprove section|date=October 2012|talk=Verifiability of definition}}
{{Calculus |Vector}}

In [[mathematics]], the '''directional derivative''' of a multivariate [[differentiable function]] along a given [[vector (mathematics)|vector]] '''v''' at a given point '''x''' intuitively represents the instantaneous rate of change of the function, moving through '''x''' with a velocity specified by '''v'''.  It therefore generalizes the notion of a [[partial derivative]], in which the rate of change is taken along one of the [[Curvilinear coordinates|curvilinear]] [[coordinate curves]], all other coordinates being constant.

The directional derivative is a special case of the [[Gâteaux derivative]].

== Definition ==
[[File:Directional derivative contour plot.svg|thumb|275px|A [[contour plot]] of <math>f(x, y)=x^2 + y^2</math>, showing the gradient vector in green, and the unit vector <math>\bold{u}</math> scaled by the directional derivative in the direction of <math>\bold{u}</math> in orange. The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function.]]

The directional derivative of a [[scalar function]] 
:<math>f(\bold{x}) = f(x_1, x_2, \ldots, x_n)</math>
along a vector
:<math>\bold{v} = (v_1, \ldots, v_n)</math>
is the [[function (mathematics)|function]] defined by the [[limit (mathematics)|limit]]<ref>{{cite book |author1=R. Wrede |author2=M.R. Spiegel | title=Advanced Calculus|edition=3rd| publisher=Schaum's Outline Series| year=2010 | isbn=978-0-07-162366-7}}</ref>
:<math>\nabla_{\bold{v}}{f}(\bold{x}) = \lim_{h \rightarrow 0}{\frac{f(\bold{x} + h\bold{v}) - f(\bold{x})}{h}}.</math>

In the context of a function on a [[Euclidean space]], some texts restrict the vector '''v''' to being a [[unit vector]].  Without the restriction, this definition is valid in a broad range of contexts, for example where the [[Euclidean norm|norm]] of a vector (and hence a unit vector) is undefined.<ref>The applicability extends to functions over spaces without a [[metric (mathematics)|metric]] and to [[differentiable manifold]]s, such as in [[general relativity]].</ref>

If the function ''f'' is [[Differentiable function#Differentiability in higher dimensions|differentiable]] at '''x''', then the directional derivative exists along any vector '''v''', and one has

:<math>\nabla_{\bold{v}}{f}(\bold{x}) = \nabla f(\bold{x}) \cdot \bold{v}</math>

where the <math>\nabla</math> on the right denotes the gradient and <math>\cdot</math> is the [[dot product]].<ref>If the dot product is undefined, the [[gradient]] is also undefined; however, for differentiable ''f'', the directional derivative is still defined, and a similar relation exists with the exterior derivative.</ref> Intuitively, the directional derivative of ''f'' at a point '''x''' represents the [[derivative|rate of change]] of ''f'', in the direction of '''v''' with respect to time, when moving past '''x'''.

=== Variation using only direction of vector for Euclidean space ===
[[image:Geometrical interpretation of a directional derivative.svg|thumb|The angle α between the tangent A and the horizontal will be maximum if the cutting plane contains the direction of the gradient A.]]
Some authors<ref>Thomas, George B. Jr.; and Finney, Ross L. (1979) ''Calculus and Analytic Geometry'', Addison-Wesley Publ. Co., fifth edition, p. 593.</ref> define the directional derivative to be with respect to an arbitrary nonzero vector '''v''' after [[Normalized vector|normalization]], thus being independent of its magnitude and depending only on its direction.<ref>This typically assumes a [[Euclidean space]] – for example, a function of several variables typically has no definition of the magnitude of a vector, and hence of a unit vector.</ref>

This definition gives the rate of increase of ''f'' per unit of distance moved in the given direction. In this case, one has
:<math>\nabla_{\bold{v}}{f}(\bold{x}) = \lim_{h \rightarrow 0}{\frac{f(\bold{x} + h\bold{v}) - f(\bold{x})}{h|\bold{v}|}},</math>
or in case ''f'' is differentiable at '''x''',
:<math>\nabla_{\bold{v}}{f}(\bold{x}) = \nabla f(\bold{x}) \cdot \frac{\bold{v}}{|\bold{v}|} .</math>

== Notation ==
Directional derivatives can be also denoted by:
:<math>\nabla_{\bold{v}}{f}(\bold{x}) \sim \frac{\partial{f(\bold{x})}}{\partial{\bold{v}}}
\sim f'_\mathbf{v}(\bold{x}) \sim D_\bold{v}f(\bold{x}) \sim Df(\bold{x})
(\bold{v}) 
\sim \partial_\bold{v}f(\bold{x}) \sim \mathbf{v}\cdot{\nabla f(\bold{x})} \sim \bold{v}\cdot \frac{\partial f(\bold{x})}{\partial\bold{x}} </math>
where ''v'' is a parameterization of a curve to which '''v''' is tangent and which determines its magnitude.

== Properties ==
Many of the familiar properties of the ordinary [[derivative]] hold for the directional derivative.  These include, for any functions ''f'' and ''g'' defined in a [[neighborhood (mathematics)|neighborhood]] of, and [[total derivative|differentiable]] at, '''p''': 
{{ordered list
|1=  '''[[Sum rule in differentiation|sum rule]]''':
:<math>\nabla_{\bold{v}} (f + g) = \nabla_{\bold{v}} f + \nabla_{\bold{v}} g</math>
|2=  '''[[Constant factor rule in differentiation|constant factor rule]]''': For any constant ''c'', 
:<math>\nabla_{\bold{v}} (cf) = c\nabla_{\bold{v}} f</math>
|3=  '''[[product rule]]''' (or '''Leibniz's rule'''):
:<math>\nabla_{\bold{v}} (fg) = g\nabla_{\bold{v}} f + f\nabla_{\bold{v}} g</math>
|4=  '''[[chain rule]]''': If ''g'' is differentiable at '''p''' and ''h'' is differentiable at ''g''('''p'''), then
:<math>\nabla_{\bold{v}}(h\circ g)(\bold{p}) = h'(g(\bold{p})) \nabla_{\bold{v}} g (\bold{p})</math>
}}

== In differential geometry ==
{{see also|Tangent space#Tangent vectors as directional derivatives}}

Let {{math|''M''}} be a [[differentiable manifold]] and {{math|'''p'''}} a point of {{math|''M''}}.  Suppose that {{math|''f''}} is a function defined in a neighborhood of {{math|'''p'''}}, and [[total derivative|differentiable]] at {{math|'''p'''}}.  If {{math|'''v'''}} is a [[tangent vector]] to {{math|''M''}} at {{math|'''p'''}}, then the '''directional derivative''' of {{math|''f''}} along {{math|'''v'''}}, denoted variously as {{math|d''f''('''v''')}} (see [[Exterior derivative]]), <math>\nabla_{\bold{v}} f(\bold{p})</math> (see [[Covariant derivative]]), <math>L_{\bold{v}} f(\bold{p})</math> (see [[Lie derivative]]), or <math>{\bold{v}}_{\bold{p}}(f)</math> (see {{section link|Tangent space|Definition via derivations}}), can be defined as follows.  Let {{math|''γ'' : [−1, 1] → ''M''}} be a differentiable curve with {{math|1=''γ''(0) = '''p'''}} and {{math|1=''γ''′(0) = '''v'''}}.  Then the directional derivative is defined by
:<math>\nabla_{\bold{v}} f(\bold{p}) = \left.\frac{d}{d\tau} f\circ\gamma(\tau)\right|_{\tau=0}</math>
This definition can be proven independent of the choice of {{math|''γ''}}, provided {{math|''γ''}} is selected in the prescribed manner so that {{math|1=''γ''′(0) = '''v'''}}.

===The Lie derivative===
The [[Lie derivative]] of a vector field <math>\scriptstyle W^\mu(x)</math> along a vector field <math>\scriptstyle V^\mu(x)</math> is given by the difference of two directional derivatives (with vanishing torsion):
:<math>\mathcal{L}_V W^\mu=(V\cdot\nabla) W^\mu-(W\cdot\nabla) V^\mu</math>
In particular, for a scalar field <math>\scriptstyle \phi(x)</math>, the Lie derivative reduces to the standard directional derivative:
:<math>\mathcal{L}_V \phi=(V\cdot\nabla) \phi</math>

===The Riemann tensor===
Directional derivatives are often used in introductory derivations of the [[Riemann curvature tensor]]. Consider a curved rectangle with an infinitesimal vector ''δ'' along one edge and ''δ''′ along the other. We translate a covector ''S'' along ''δ'' then ''δ''′ and then subtract the translation along ''δ''′ and then ''δ''. Instead of building the directional derivative using partial derivatives, we use the [[covariant derivative]]. The translation operator for ''δ'' is thus
:<math>1+\sum_\nu \delta^\nu D_\nu=1+\delta\cdot D</math>
and for ''δ''′
:<math>1+\sum_\mu \delta'^\mu D_\mu=1+\delta'\cdot D</math>
The difference between the two paths is then
:<math>(1+\delta'\cdot D)(1+\delta\cdot D)S^\rho-(1+\delta\cdot D)(1+\delta'\cdot D)S^\rho=\sum_{\mu,\nu}\delta'^\mu \delta^\nu[D_\mu,D_\nu]S_\rho</math>
It can be argued<ref>{{cite book|last1=Zee|first1=A.|title=Einstein gravity in a nutshell|date=2013|publisher=Princeton University Press|location=Princeton|isbn=9780691145587|page=341}}</ref> that the noncommutativity of the covariant derivatives measures the curvature of the manifold:
:<math>[D_\mu,D_\nu]S_\rho=\pm \sum_\sigma R^\sigma_{\rho\mu\nu}S_\sigma</math>
with R the Riemann tensor of course and the sign depending on the [[sign convention]] of the author.

== In group theory ==

===Translations===
In the [[Poincaré algebra]], we can define an infinitesimal translation operator '''P''' as
:<math>\mathbf{P}=i\nabla</math>
(the i ensures that '''P''' is a [[self-adjoint operator]]) For a finite displacement '''λ''', the [[Unitary operator|unitary]] [[Hilbert space]] [[Group representation|representation]] for translations is<ref>{{cite book|last1=Weinberg|first1=Steven|title=The quantum theory of fields|date=1999|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=9780521550017|edition=Reprinted (with corr.).}}</ref> 
:<math>U(\boldsymbol{\lambda})=\exp\left(-i\boldsymbol{\lambda}\cdot\mathbf{P}\right)</math>
By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative:
:<math>U(\boldsymbol{\lambda})=\exp\left(\boldsymbol{\lambda}\cdot\nabla\right)</math>
This is a translation operator in the sense that it acts on multivariable functions f('''x''') as
:<math>U(\boldsymbol{\lambda}) f(\mathbf{x})=\exp\left(\boldsymbol{\lambda}\cdot\nabla\right) f(\mathbf{x})=f(\mathbf{x}+\boldsymbol{\lambda}) </math>

{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Proof of the last equation
|-
|
In standard single-variable calculus, the derivative of a smooth function f(x) is defined by (for small ε)
:<math>\frac{df}{dx}=\frac{f(x+\epsilon)-f(x)}{\epsilon}</math>
This can be rearranged to find f(x+ε):
:<math>f(x+\epsilon)=f(x)+\epsilon \,\frac{df}{dx}=\left(1+\epsilon\,\frac{d}{dx}\right)f(x)</math>
It follows that <math>[1+\epsilon\,(d/dx)] </math> is a translation operator. This is instantly generalized<ref>{{cite book|last1=Zee|first1=A.|title=Einstein gravity in a nutshell|date=2013|publisher=Princeton University Press|location=Princeton|isbn=9780691145587}}</ref> to multivariable functions f('''x''')
:<math>f(\mathbf{x}+\boldsymbol{\epsilon})=\left(1+\boldsymbol{\epsilon}\cdot\nabla\right)f(\mathbf{x})</math>
Here <math> \boldsymbol{\epsilon}\cdot\nabla</math> is the directional derivative along the infinitesimal displacement '''ε'''. We have found the infinitesimal version of the translation operator:
:<math>U(\boldsymbol{\epsilon})=1+\boldsymbol{\epsilon}\cdot\nabla</math>
It is evident that the group multiplication law<ref>{{cite book|last1=Mexico|first1=Kevin Cahill, University of New|title=Physical mathematics|date=2013|publisher=Cambridge University Press|location=Cambridge|isbn=978-1107005211|edition=Repr.}}</ref> U(g)U(f)=U(gf) takes the form
:<math>U(\mathbf{a})U(\mathbf{b})=U(\mathbf{a+b})</math>
So suppose that we take the finite displacement '''λ''' and divide it into N parts (N→∞ is implied everywhere), so that '''λ'''/N='''ε'''. In other words,
:<math>\boldsymbol{\lambda}=N\boldsymbol{\epsilon}</math>
Then by applying U('''ε''') N times, we can construct U('''λ'''):
:<math>[U(\boldsymbol{\epsilon})]^N=U(N\boldsymbol{\epsilon})=U(\boldsymbol{\lambda})</math>
We can now plug in our above expression for U('''ε'''):
:<math>[U(\boldsymbol{\epsilon})]^N=\left[1+\boldsymbol{\epsilon}\cdot\nabla\right]^N=\left[1+\frac{\boldsymbol{\lambda}\cdot\nabla}{N}\right]^N</math>
Using the identity<ref>{{cite book|last1=Edwards|first1=Ron Larson, Robert, Bruce H.|title=Calculus of a single variable|date=2010|publisher=Brooks/Cole|location=Belmont|isbn=9780547209982|edition=9th}}</ref>
:<math>\exp(x)=\left[1+\frac{x}{N}\right]^N</math>
we have
:<math>U(\boldsymbol{\lambda})=\exp\left(\boldsymbol{\lambda}\cdot\nabla\right)</math>
And since U('''ε''')f('''x''')=f('''x'''+'''ε''') we have
:<math>[U(\boldsymbol{\epsilon})]^Nf(\mathbf{x})=f(\mathbf{x}+N\boldsymbol{\epsilon})=f(\mathbf{x}+\boldsymbol{\lambda})=U(\boldsymbol{\lambda})f(\mathbf{x})=\exp\left(\boldsymbol{\lambda}\cdot\nabla\right)f(\mathbf{x})</math>
Q.E.D.

As a technical note, this procedure is only possible because the translation group forms an [[abelian group|Abelian]] [[subgroup]] ([[Cartan subalgebra]]) in the Poincaré algebra. In particular, the group multiplication law U('''a''')U('''b''')=U('''a'''+'''b''') should not be taken for granted. We also note that Poincaré is a connected Lie group. It is a group of transformations  T(ξ) that are described by a continuous set of real parameters <math>\scriptstyle \xi^a</math>. The group multiplication law takes the form
:<math>T(\bar{\xi})T(\xi)=T(f(\bar{\xi},\xi))</math>
Taking <math>\scriptstyle \xi^a</math>=0 as the coordinates of the identity, we must have
:<math>f^a(\xi,0)=f^a(0,\xi)=\xi^a</math>
The actual operators on the Hilbert space are represented by unitary operators U(T(ξ)). In the above notation we suppressed the T; we now write U('''λ''') as U('''P'''('''λ''')). For a small neighborhood around the identity, the power series representation 
:<math>U(T(\xi))=1+i\sum_a\xi^a t_a+\frac{1}{2}\sum_{b,c}\xi^b\xi^c t_{bc}+\cdots</math>
is quite good. Suppose that U(T(ξ)) form a non-projective representation, i.e. that
:<math>U(T(\bar{\xi}))U(T(\xi))=U(T(f(\bar{\xi},\xi)))</math>
The expansion of f to second power is
:<math>f^a(\bar{\xi},\xi)=\xi^a+\bar{\xi}^a+\sum_{b,c}f^{abc}\bar{\xi}^b\xi^c</math>
After expanding the representation multiplication equation and equating coefficients, we have the nontrivial condition
:<math>t_{bc}=-t_b t_c-i\sum_a f^{abc}t_a</math>
Since <math>\scriptstyle t_{ab}</math> is by definition symmetric in its indices, we have the standard [[Lie algebra]] commutator:
:<math>[t_b, t_c]=i\sum_a(-f^{abc}+f^{acb})t_a=i\sum_a C^{abc}t_a</math>
with C the [[structure constant]]. The generators for translations are partial derivative operators, which commute:
:<math>\left[\frac{\partial}{\partial x^b},\frac{\partial }{\partial x^c}\right]=0 </math>
This implies that the structure constants vanish and thus the quadratic coefficients in the f expansion vanish as well. This means that f is simply additive:
:<math>f^a_\text{abelian}(\bar{\xi},\xi)=\xi^a+\bar{\xi}^a</math>
and thus for abelian groups,
:<math>U(T(\bar{\xi}))U(T(\xi))=U(T(\bar{\xi}+\xi))</math>
Q.E.D.
|}

===Rotations===
The [[rotation operator (quantum mechanics)|rotation operator]] also contains a directional derivative. The rotation operator for an angle '''θ''', i.e. by an amount θ=|'''θ'''| about an axis parallel to <math>\scriptstyle \hat{\theta}</math>='''θ'''/θ is
::<math>U(R(\boldsymbol{\theta}))=\exp(-i\boldsymbol{\theta}\cdot\mathbf{L})</math>
Here '''L''' is the vector operator that generates [[SO(3)]]:
:<math>\mathbf{L}=\begin{pmatrix}
 0& 0 & 0\\ 
 0& 0 & 1\\ 
 0& -1 & 0
\end{pmatrix}\mathbf{i}+\begin{pmatrix}
0 &0  & -1\\ 
 0& 0 &0 \\ 
1 & 0 & 0
\end{pmatrix}\mathbf{j}+\begin{pmatrix}
 0&1  &0 \\ 
 -1&0  &0 \\ 
0 & 0 & 0
\end{pmatrix}\mathbf{k}</math>
It may be shown geometrically that an infinitesimal right-handed rotation changes the position vector '''x''' by
:<math>\mathbf{x}\rightarrow \mathbf{x}-\delta\boldsymbol{\theta}\times\mathbf{x}</math>
So we would expect under infinitesimal rotation:
:<math>U(R(\delta\boldsymbol{\theta}))f(\mathbf{x})=f(\mathbf{x}-\delta\boldsymbol{\theta}\times\mathbf{x})=f(\mathbf{x})-(\delta\boldsymbol{\theta}\times\mathbf{x})\cdot\nabla f</math>
It follows that 
:<math>U(R(\delta\boldsymbol{\theta}))=1-(\delta\boldsymbol{\theta}\times\mathbf{x})\cdot\nabla</math>
Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative:<ref>{{cite book|last1=Shankar|first1=R.|title=Principles of quantum mechanics|date=1994|publisher=Kluwer Academic / Plenum|location=New York|isbn=9780306447907|page=318|edition=2nd}}</ref>
:<math>U(R(\boldsymbol{\theta}))=\exp(-(\boldsymbol{\theta}\times\mathbf{x})\cdot\nabla)</math>

== Normal derivative ==

A '''normal derivative''' is a directional derivative taken in the direction normal (that is, [[orthogonal]]) to some surface in space, or more generally along a [[normal vector]] field orthogonal to some [[hypersurface]]. See for example [[Neumann boundary condition]].  If the normal direction is denoted by <math>\bold{n}</math>, then the directional derivative of a function ''f'' is sometimes denoted as <math>\frac{ \partial f}{\partial n}</math>.  In other notations
:<math>\frac{ \partial f}{\partial n} = \nabla f(\bold{x}) \cdot \bold{n} = \nabla_{\bold{n}}{f}(\bold{x}) = \frac{\partial f}{\partial \bold{x}}\cdot\bold{n} = Df(\bold{x})[\bold{n}] </math>

== In the continuum mechanics of solids ==

Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and of [[tensors]] with respect to vectors and tensors.<ref name=Marsden00>J. E. Marsden and T. J. R. Hughes, 2000, ''Mathematical Foundations of Elasticity'', Dover.</ref>  The '''directional directive''' provides a systematic way of finding these derivatives.

The definitions of directional derivatives for various situations are given below.  It is assumed that the functions are sufficiently smooth that derivatives can be taken.

===Derivatives of scalar valued functions of vectors===
Let <math>f(\mathbf{v})</math> be a real valued function of the vector <math>\mathbf{v}</math>.  Then the derivative of <math>f(\mathbf{v})</math> with respect to <math>\mathbf{v}</math> (or at <math>\mathbf{v}</math>) in the direction <math>\mathbf{u}</math> is defined as
:<math>
  \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = Df(\mathbf{v})[\mathbf{u}] 
     = \left[\frac{d }{d \alpha}~f(\mathbf{v} + \alpha~\mathbf{u})\right]_{\alpha = 0}
</math>
for all vectors <math>\mathbf{u}</math>.

''Properties:''
{{ordered list
|1= If <math>f(\mathbf{v}) = f_1(\mathbf{v}) + f_2(\mathbf{v})</math> then <math>
   \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial f_1}{\partial \mathbf{v}} + \frac{\partial f_2}{\partial \mathbf{v}}\right)\cdot\mathbf{u} 
 </math>

|2= If <math>f(\mathbf{v}) = f_1(\mathbf{v})~ f_2(\mathbf{v})</math> then <math>
   \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial f_1}{\partial \mathbf{v}}\cdot\mathbf{u}\right)~f_2(\mathbf{v}) + f_1(\mathbf{v})~\left(\frac{\partial f_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)
 </math>

|3= If <math>f(\mathbf{v}) = f_1(f_2(\mathbf{v}))</math> then <math>
   \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} =  \frac{\partial f_1}{\partial f_2}~\frac{\partial f_2}{\partial \mathbf{v}}\cdot\mathbf{u} 
 </math>
}}

===Derivatives of vector valued functions of vectors===
Let <math>\mathbf{f}(\mathbf{v})</math> be a vector valued function of the vector <math>\mathbf{v}</math>.  Then the derivative of <math>\mathbf{f}(\mathbf{v})</math> with respect to <math>\mathbf{v}</math> (or at <math>\mathbf{v}</math>) in the direction <math>\mathbf{u}</math> is the '''second-order tensor''' defined as
:<math>
  \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = D\mathbf{f}(\mathbf{v})[\mathbf{u}] 
     = \left[\frac{d }{d \alpha}~\mathbf{f}(\mathbf{v} + \alpha~\mathbf{u})\right]_{\alpha = 0}
</math>
for all vectors <math>\mathbf{u}</math>.

''Properties:''
{{ordered list
|1= If <math>\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{v}) + \mathbf{f}_2(\mathbf{v})</math> then <math>
   \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial \mathbf{f}_1}{\partial \mathbf{v}} + \frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\right)\cdot\mathbf{u} 
 </math>

|2= If <math>\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{v})\times\mathbf{f}_2(\mathbf{v})</math> then <math>
   \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial \mathbf{f}_1}{\partial \mathbf{v}}\cdot\mathbf{u}\right)\times\mathbf{f}_2(\mathbf{v}) + \mathbf{f}_1(\mathbf{v})\times\left(\frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)
 </math>

|3= If <math>\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{f}_2(\mathbf{v}))</math> then <math>
   \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} =  \frac{\partial \mathbf{f}_1}{\partial \mathbf{f}_2}\cdot\left(\frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)
 </math>
}}

===Derivatives of scalar valued functions of second-order tensors===
Let <math>f(\boldsymbol{S})</math> be a real valued function of the second order tensor <math>\boldsymbol{S}</math>.  Then the derivative of <math>f(\boldsymbol{S})</math> with respect to <math>\boldsymbol{S}</math> (or at <math>\boldsymbol{S}</math>) in the direction
<math>\boldsymbol{T}</math> is the ''' second order tensor''' defined as
:<math>
  \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = Df(\boldsymbol{S})[\boldsymbol{T}] 
     = \left[\frac{d }{d \alpha}~f(\boldsymbol{S} + \alpha\boldsymbol{T})\right]_{\alpha = 0}
</math>
for all second order tensors <math>\boldsymbol{T}</math>.

''Properties:''
{{ordered list
|1=  If <math>f(\boldsymbol{S}) = f_1(\boldsymbol{S}) + f_2(\boldsymbol{S})</math> then <math> \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \left(\frac{\partial f_1}{\partial \boldsymbol{S}} + \frac{\partial f_2}{\partial \boldsymbol{S}}\right):\boldsymbol{T} </math>

|2= If <math>f(\boldsymbol{S}) = f_1(\boldsymbol{S})~ f_2(\boldsymbol{S})</math> then <math> \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \left(\frac{\partial f_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)~f_2(\boldsymbol{S}) + f_1(\boldsymbol{S})~\left(\frac{\partial f_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) </math>

|3= If <math>f(\boldsymbol{S}) = f_1(f_2(\boldsymbol{S}))</math> then <math> \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \frac{\partial f_1}{\partial f_2}~\left(\frac{\partial f_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) </math>
}}

===Derivatives of tensor valued functions of second-order tensors===
Let <math>\boldsymbol{F}(\boldsymbol{S})</math> be a second order tensor valued function of the second order tensor <math>\boldsymbol{S}</math>.  Then the derivative of <math>\boldsymbol{F}(\boldsymbol{S})</math> with respect to <math>\boldsymbol{S}</math> 
(or at <math>\boldsymbol{S}</math>) in the direction <math>\boldsymbol{T}</math> is the ''' fourth order tensor''' defined as
:<math>
  \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = D\boldsymbol{F}(\boldsymbol{S})[\boldsymbol{T}] 
     = \left[\frac{d }{d \alpha}~\boldsymbol{F}(\boldsymbol{S} + \alpha\boldsymbol{T})\right]_{\alpha = 0}
</math>
for all second order tensors <math>\boldsymbol{T}</math>.

''Properties:''
{{ordered list
|1= If <math>\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{S}) + \boldsymbol{F}_2(\boldsymbol{S})</math> then <math> \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}} + \frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}\right):\boldsymbol{T} </math>

|2= If <math>\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{S})\cdot\boldsymbol{F}_2(\boldsymbol{S})</math> then <math> \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)\cdot\boldsymbol{F}_2(\boldsymbol{S}) + \boldsymbol{F}_1(\boldsymbol{S})\cdot\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) </math>

|3= If <math>\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{F}_2(\boldsymbol{S}))</math> then <math> \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{F}_2}:\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) </math>

|4= If <math>f(\boldsymbol{S}) = f_1(\boldsymbol{F}_2(\boldsymbol{S}))</math> then <math> \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} =  \frac{\partial f_1}{\partial \boldsymbol{F}_2}:\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) </math>
}}

== See also ==
* [[Fréchet derivative]]
* [[Gâteaux derivative]]
* [[Derivative (generalizations)]]
* [[Lie derivative]]
* [[Differential form]]
* [[Structure tensor]]
* [[Tensor derivative (continuum mechanics)]]
* [[Del in cylindrical and spherical coordinates]]

== Notes ==
{{reflist}}

== References ==
*{{cite book | first=F. B. | last=Hildebrand | title=Advanced Calculus for Applications| publisher=Prentice Hall | year=1976 | isbn=0-13-011189-9 }}
*{{cite book |author1=K.F. Riley |author2=M.P. Hobson |author3=S.J. Bence | title=Mathematical methods for physics and engineering| publisher=Cambridge University Press| year=2010 | isbn=978-0-521-86153-3}}

== External links ==
*[http://mathworld.wolfram.com/DirectionalDerivative.html Directional derivatives] at [[MathWorld]].
*[http://planetmath.org/directionalderivative Directional derivative] at [[PlanetMath]].

[[Category:Differential calculus]]
[[Category:Differential geometry]]
[[Category:Generalizations of the derivative]]
[[Category:Multivariable calculus]]