Editing Partial derivative (section)

==Notation==
{{see|∂}}
For the following examples, let {{mvar|f}} be a function in {{mvar|x}}, {{mvar|y}}, and {{mvar|z}}.

First-order partial derivatives:

<math display="block">\frac{ \partial f}{ \partial x} = f'_x = \partial_x f.</math>

Second-order partial derivatives:

<math display="block">\frac{ \partial^2 f}{ \partial x^2} = f''_{xx} = \partial_{xx} f = \partial_x^2 f.</math>

Second-order [[mixed derivatives]]:

<math display="block">\frac{\partial^2 f}{\partial y \,\partial x} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = (f'_{x})'_{y} = f''_{xy} = \partial_{yx} f = \partial_y \partial_x f .</math>

Higher-order partial and mixed derivatives:

<math display="block">\frac{\partial^{i+j+k} f}{\partial x^i \partial y^j \partial z^k } = f^{(i, j, k)} = \partial_x^i \partial_y^j \partial_z^k f.</math>

When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as [[statistical mechanics]], the partial derivative of {{mvar|f}} with respect to {{mvar|x}}, holding {{mvar|y}} and {{mvar|z}} constant, is often expressed as

<math display="block">\left( \frac{\partial f}{\partial x} \right)_{y,z} .</math>

Conventionally, for clarity and simplicity of notation, the partial derivative ''function'' and the ''value'' of the function at a specific point are [[Abuse of notation|conflated]] by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like 

<math display="block">\frac{\partial f(x,y,z)}{\partial x}</math> 

is used for the function, while 

<math display="block">\frac{\partial f(u,v,w)}{\partial u}</math> 

might be used for the value of the function at the point {{nowrap|<math>(x,y,z)=(u,v,w)</math>.}} However, this convention breaks down when we want to evaluate the partial derivative at a point like {{nowrap|<math>(x,y,z)=(17, u+v, v^2)</math>.}} In such a case, evaluation of the function must be expressed in an unwieldy manner as 

<math display="block">\frac{\partial f(x,y,z)}{\partial x}(17, u+v, v^2)</math> 

or 

<math display="block">\left. \frac{\partial f(x,y,z)}{\partial x}\right |_{(x,y,z)=(17, u+v, v^2)}</math> 

in order to use the Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with <math>D_i</math> as the partial derivative symbol with respect to the {{mvar|i}}-th variable. For instance, one would write <math>D_1 f(17, u+v, v^2)</math> for the example described above, while the expression <math>D_1 f</math> represents the partial derivative ''function'' with respect to the first variable.<ref>{{Cite book |last= Spivak |first= M. |date= 1965 |title= Calculus on Manifolds |publisher= W. A. Benjamin |location= New York |pages= 44 |isbn= 9780805390216 |url=https://archive.org/details/SpivakM.CalculusOnManifoldsPerseus2006Reprint }}</ref>

For higher order partial derivatives, the partial derivative (function) of <math>D_i f</math> with respect to the {{mvar|j}}-th variable is denoted {{nowrap|<math>D_j(D_i f)=D_{i,j} f</math>.}} That is, {{nowrap|<math>D_j\circ D_i =D_{i,j}</math>,}} so that the variables are listed in the order in which the derivatives are taken, and thus, in reverse order of how the composition of operators is usually notated. Of course, [[Clairaut's theorem on equality of mixed partials|Clairaut's theorem]] implies that <math>D_{i,j}=D_{j,i}</math> as long as comparatively mild regularity conditions on {{mvar|f}} are satisfied.