Introduction: In Chapter 4: Differentiation of functions of two variables the partial derivatives of functions of two variables are discussed. Since a function $z(x,y)$ has two partial derivatives, $z'_x(x,y)$ and $z'_y(x,y)$, there are four second-order partial derivatives.
Definition: Let $z(x,y)$ be a function of two variables. Then
Remark: For all the functions discussed in this book it holds that $z''_{xy}(x,y)=z''_{yx}(x,y)$.
Definition: Let $z(x,y)$ be a function of two variables. Then
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$z''_{xx}(x,y)$ is the derivative with respect to $x$ of $z'_x(x,y)$,
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$z''_{yx}(x,y)$ is the derivative with respect to $y$ of $z'_x(x,y)$,
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$z''_{xy}(x,y)$ is the derivative with respect to $x$ of $z'_y(x,y)$,
- $z''_{yy}(x,y)$ is the derivative with respect to $y$ of $z'_y(x,y)$.
Remark: For all the functions discussed in this book it holds that $z''_{xy}(x,y)=z''_{yx}(x,y)$.