Editing Partial derivative (section)

==Higher order partial derivatives==

Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function <math>f(x, y, ...)</math> the "own" second partial derivative with respect to {{mvar|x}} is simply the partial derivative of the partial derivative (both with respect to {{mvar|x}}):<ref>{{cite book |last= Chiang |first= Alpha C. |date= 1984 |title= Fundamental Methods of Mathematical Economics |publisher= McGraw-Hill |edition= 3rd |author-link= Alpha Chiang }}</ref>{{rp|316–318}}

<math display="block">\frac{\partial ^2 f}{\partial x^2} \equiv \partial \frac{{\partial f / \partial x}}{{\partial x}} \equiv \frac{{\partial f_x }}{{\partial x }} \equiv f_{xx}.</math>

The cross partial derivative with respect to {{mvar|x}} and {{mvar|y}} is obtained by taking the partial derivative of {{mvar|f}} with respect to {{mvar|x}}, and then taking the partial derivative of the result with respect to {{mvar|y}}, to obtain

<math display="block">\frac{\partial ^2 f}{\partial y\, \partial x} \equiv \partial \frac{\partial f / \partial x}{\partial y} \equiv \frac{\partial f_x}{\partial y} \equiv f_{xy}.</math>

[[Schwarz theorem|Schwarz's theorem]] states that if the second derivatives are continuous, the expression for the cross partial derivative is unaffected by which variable the partial derivative is taken with respect to first and which is taken second. That is,

<math display="block">\frac {\partial ^2 f}{\partial x\, \partial y} = \frac{\partial ^2 f}{\partial y\, \partial x}</math>

or equivalently <math>f_{yx} = f_{xy}.</math>

Own and cross partial derivatives appear in the [[Hessian matrix]] which is used in the [[second order condition]]s in [[optimization]] problems.
The higher order partial derivatives can be obtained by successive differentiation