Step By Step Calculus » 11.4 - Higher Order Partial Derivatives

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Synopsis
Given a function, f(x,y)f(x,y), and assuming that differentiation produces functions where necessary, the following are all multiple-order derivatives of f(x,y)f(x,y):
  • \dfrac{\partial^2}{\partial x^2}f(x,y)\dfrac{\partial^2}{\partial x^2}f(x,y), \dfrac{\partial^2}{\partial x\partial y}f(x,y)\dfrac{\partial^2}{\partial x\partial y}f(x,y), and \dfrac{\partial^2}{\partial y\partial x}f(x,y)\dfrac{\partial^2}{\partial y\partial x}f(x,y) are all second-order partial derivatives of ff.
  • \dfrac{\partial^3}{\partial x^3}f(x,y)\dfrac{\partial^3}{\partial x^3}f(x,y), \dfrac{\partial^3}{\partial x^2\partial y}f(x,y)\dfrac{\partial^3}{\partial x^2\partial y}f(x,y), and \dfrac{\partial^3}{\partial y^2\partial x}f(x,y)\dfrac{\partial^3}{\partial y^2\partial x}f(x,y) are all third-order partial derivatives of ff.
When evaluating multiple-order partial derivatives, it is common practice to begin at the innermost partial, and work your way out, such that \dfrac{\partial^2}{\partial x \partial y }f(x,y)=\dfrac{\partial}{\partial x}\left(\dfrac{\partial}{\partial y}f(x,y)\right)\dfrac{\partial^2}{\partial x \partial y }f(x,y)=\dfrac{\partial}{\partial x}\left(\dfrac{\partial}{\partial y}f(x,y)\right).
However, it is sometimes much more convenient to reorder the order of differentiation, which we can do if the end result is continuous. By the Clairaut’s Theorem, if \dfrac{\partial }{\partial y}\dfrac{\partial }{\partial x}f(x,y)\dfrac{\partial }{\partial y}\dfrac{\partial }{\partial x}f(x,y) is continuous, then \dfrac{\partial }{\partial y}\dfrac{\partial }{\partial x}f(x,y)=\dfrac{\partial }{\partial x}\dfrac{\partial }{\partial y}f(x,y)\dfrac{\partial }{\partial y}\dfrac{\partial }{\partial x}f(x,y)=\dfrac{\partial }{\partial x}\dfrac{\partial }{\partial y}f(x,y).