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.