Second-order partial derivatives

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

• $z''_{xx}(x,y)$ is the derivative with respect to $x$ of $z'_x(x,y)$,
  
• $z''_{yx}(x,y)$ is the derivative with respect to $y$ of $z'_x(x,y)$,
  
• $z''_{xy}(x,y)$ 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 $z''_{xy}(x,y)=z''_{yx}(x,y)$.