Second-Order Partial Derivatives

Second-Order Partial Derivatives. Assume that \(z=f(x,y)\text{.}\)

• The second partial derivative with respect to \(x\text{:}\)

  \begin{equation*} f_{xx} =(f_{x})_{x} =\frac{\partial^2 z}{\partial x^2}. \end{equation*}

• The second partial derivative with respect to \(y\text{:}\)

  \begin{equation*} f_{yy} =(f_{y})_{y} =\frac{\partial^2 z}{\partial y^2}. \end{equation*}

• The mixed partial derivative with respect to \(x\) and then \(y\text{:}\)

  \begin{equation*} f_{xy} =(f_{x})_{y} =\frac{\partial^2 z}{\partial y \partial x}. \end{equation*}

• The mixed partial derivative with respect to \(y\) and then \(x\text{:}\)

  \begin{equation*} f_{yx} =(f_{y})_{x} =\frac{\partial^2 z}{\partial x \partial y}. \end{equation*}

Equality of Mixed Partial Derivatives: If \(f_{xy}\) and \(f_{yx}\) are continuous at \((a,b)\text{,}\) then \(f_{xy}(a,b)=f_{yx}(a,b)\text{.}\)

Higher-order partial derivatives such as the third and fourth derivatives can be defined similarly.

Higher-order partial derivatives can also be defined for functions of more than two variables.

Second-order derivatives measure concavity, or how slope changes. Specifically:

• \(f_{xx}\text{:}\) positive if the slope \(f_x\) is increasing as we move in the \(x\)-direction; negative if the slope \(f_x\) is decreasing as we move in the \(x\)-direction.

• \(f_{yy}\text{:}\) positive if the slope \(f_y\) is increasing as we move in the \(y\)-direction; negative if the slope \(f_y\) is decreasing as we move in the \(y\)-direction.

• \(f_{xy}\text{:}\) positive if the slope \(f_x\) in the \(x\)-direction is increasing as we move in the \(y\)-direction; negative if the slope \(f_x\) in the \(x\)-direction is decreasing as we move in the \(y\)-direction.

• \(f_{yx}\text{:}\) positive if the slope \(f_y\) in the \(y\)-direction is increasing as we move in the \(x\)-direction; negative if the slope \(f_y\) in the \(y\)-direction is decreasing as we move in the \(x\)-direction.

The sign of a second derivative at a point can be determined from a contour diagram if you know the signs of \(f_x\) and \(f_y\) and how the spacing of contours are changing.