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, zx(x,y) and zy(x,y), there are four second-order partial derivatives.

Definition: Let z(x,y) be a function of two variables. Then
  • zxx(x,y) is the derivative with respect to x of zx(x,y),
  • zyx(x,y) is the derivative with respect to y of zx(x,y),
  • zxy(x,y) is the derivative with respect to x of zy(x,y),
  • zyy(x,y) is the derivative with respect to y of zy(x,y).

Remark: For all the functions discussed in this book it holds that zxy(x,y)=zyx(x,y).