Partial Derivatives

Functions of Two Variables

A function of two variables is usually written as $z = f(x,y)$ , where $x$ and $y$ are the independent variables and $z$ is the dependent variable. The domain of the function is the set of ordered pairs $(x,y)$ and the set of corresponding values of $z$ is called the range.

If $z=f(x,y)$ , then the partial derivatives with respect to $x$ and $y$ will be

(1) $\begin{equation*}\frac{\partial z}{\partial x} = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y) - f(x,y)}{\Delta x} ,\end{equation*}$

and

(2) $\begin{equation*}\frac{\partial z}{\partial y} = \lim_{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y) - f(x,y)}{\Delta y} ,\end{equation*}$

respectively. In computing $\frac{\partial z}{\partial x}$ , we will use the laws of ordinary differentiation while treating $y$ as a constant. Similarly, in computing $\frac{\partial z}{\partial y}$ , we will use the laws of ordinary differentiation while treating $x$ as a constant.

Alternative Symbols

If $z=f(x,y)$ , the partial derivatives can be represented using the following alternative symbols

(3) $\begin{equation*}\frac{\partial z}{\partial x} = \frac{\partial f}{\partial x} = z_x = f_x ,\end{equation*}$

and

(4) $\begin{equation*}\frac{\partial z}{\partial y} = \frac{\partial f}{\partial y} = z_y = f_y .\end{equation*}$

High-Order and Mixed Derivatives and the Corresponding Alternative Symbols

If $z=f(x,y)$ , the second-order partial derivative with respect to $x$ will be

(5) $\begin{equation*}f_{xx} = z_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial^2 z}{\partial x^2} = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x} \right) = \frac{\partial}{\partial x} \left( \frac{\partial z}{\partial x} \right) .\end{equation*}$

The third-order partial derivative with respect to $y$ will be

(6) $\begin{equation*}f_{yyy} = z_{yyy} = \frac{\partial^3 f}{\partial y^3} = \frac{\partial^3 z}{\partial y^3} = \frac{\partial}{\partial y} \left( \frac{\partial^2 f}{\partial y^2} \right) = \frac{\partial}{\partial y} \left( \frac{\partial^2 z}{\partial y^2} \right) .\end{equation*}$

And, the mixed second-order partial derivative is

(7) $\begin{equation*}f_{xy} = z_{xy} = \frac{\partial^2 f}{\partial xy} = \frac{\partial^2 z}{\partial xy} = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) = \frac{\partial}{\partial x} \left( \frac{\partial z}{\partial y} \right) .\end{equation*}$

Chain Rule

Let’s say that $z=f(u,v)$ , and $u=g(x,y)$ and $v=h(x,y)$ have continuous first-derivatives, then the chain rule is expressed as

(8) $\begin{equation*}\frac{\partial z}{\partial x} = \frac{\partial z}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial z}{\partial v} \frac{\partial v}{\partial x} ,\end{equation*}$

and/or

(9) $\begin{equation*}\frac{\partial z}{\partial y} = \frac{\partial z}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial z}{\partial v} \frac{\partial v}{\partial y} .\end{equation*}$

8-Minute Lecture on Partial Derivatives

Reference

Dennis G. Zill. Advanced Engineering Mathematics, $6^{th}$ edition. Jones $&$ Bartlett Learning. 2016.