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2.12 Higher order partial derivatives

Higher-order partial derivatives are formed by repeated differentiation. We define

∂2⁡f∂⁡x2=∂∂⁡x⁢(∂⁡f∂⁡x)=fx⁢x,  ∂2⁡f∂⁡y2=∂∂⁡y⁢(∂⁡f∂⁡y)=fy⁢y{{\partial^{2}f}\over{\partial x^{2}}}={{\partial}\over{\partial x}}\Bigl({{% \partial f}\over{\partial x}}\Bigr)=f_{xx},\qquad{{\partial^{2}f}\over{% \partial y^{2}}}={{\partial}\over{\partial y}}\Bigl({{\partial f}\over{% \partial y}}\Bigr)=f_{yy}

and the mixed partial derivatives

∂2⁡f∂⁡x⁢∂⁡y=∂∂⁡x⁢(∂⁡f∂⁡y)=fy⁢x,  ∂2⁡f∂⁡y⁢∂⁡x=∂∂⁡y⁢(∂⁡f∂⁡x)=fx⁢y{{\partial^{2}f}\over{\partial x\partial y}}={{\partial}\over{\partial x}}% \Bigl({{\partial f}\over{\partial y}}\Bigr)=f_{yx},\qquad{{\partial^{2}f}\over% {\partial y\partial x}}={{\partial}\over{\partial y}}\Bigl({{\partial f}\over{% \partial x}}\Bigr)=f_{xy}

where these exist. We think of the notation fx⁢yf_{xy} as an abbreviation for (fx)y(f_{x})_{y} so that the xx partial derivative is carried out before the yy partial derivative. This convention is not followed in all books, but fortunately the difference is inconsequential on account of the theorem below. In general, the order is the total number of times that a function has been differentiated in all variables.