\begin{definition}[Definition:Partial Derivative/Second Derivative]
Let $\map f {x, y}$ be a function of the two independent variables $x$ and $y$.
The '''second partial derivatives of $f$ with respect to $x$ and $y$''' are defined and denoted by:
{{begin-eqn}}
{{eqn | n = 1
      | l = \dfrac {\partial^2 f} {\partial x^2}
      | r = \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial x} }
      | rr= =: \map {f_{1 1} } {x, y}
      | c = 
}}
{{eqn | n = 2
      | l = \dfrac {\partial^2 f} {\partial y^2}
      | r = \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial y} }
      | rr= =: \map {f_{2 2} } {x, y}
      | c = 
}}
{{eqn | n = 3
      | l = \quad \dfrac {\partial^2 f} {\partial x \partial y}
      | r = \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial y} }
      | rr= =: \map {f_{2 1} } {x, y}
      | c = 
}}
{{eqn | n = 4
      | l = \dfrac {\partial^2 f} {\partial y \partial x}
      | r = \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial x} }
      | rr= =: \map {f_{1 2} } {x, y}
      | c = 
}}
{{end-eqn}}
\end{definition}