### Multivariate functions: Partial derivatives

### Higher partial derivatives

Higher partial derivatives arise by repeatedly taking partial derivatives.

Partial derivatives of higher order

The partial derivatives of a function of #m# variables are themselves again functions of #m# variables. In particular, you can take their partial derivatives again. These are called **partial derivatives of the second order**.

In general, the **partial derivatives of the function** #f(x_1,x_2,\ldots,x_m)# **of the #k#th order** are the functions of the form \[\frac{\partial}{\partial x_{i_1}}\frac{\partial}{\partial x_{i_2}}\cdots \frac{\partial}{\partial x_{i_k}}f(x_1,x_2,\ldots,x_m)\tiny,\] where #i_1,i_2,\ldots,i_k# are integers between #1# and #m#.

For example, the partial derivatives of \(f(x,y)\) of the second order are equal to \[ \begin{array}{rcl}\frac{\partial}{\partial x}\!\!\left(\frac{\partial}{\partial x}\! f(x,y)\right) &,& \frac{\partial}{\partial y}\!\!\left(\frac{\partial}{\partial x}\! f(x,y)\right) ,\\\frac{\partial}{\partial x}\!\!\left(\frac{\partial}{\partial y}\! f(x,y)\right)&,& \frac{\partial}{\partial y}\!\!\left(\frac{\partial}{\partial y}\! f(x,y)\right) \tiny.\end{array}\]

The partial derivatives of the following function of two variables \[f(x,y)=\frac{x}{x+y}\] can be calculated by use of the quotient rule: \[ \begin{array}{rcl} \frac{\partial}{\partial x}\!\!\left(\frac{x}{x+y}\right) = \frac{y}{(x+y)^2}&\text{ and }&\frac{\partial}{\partial y}\!\!\left(\frac{x}{x+y}\right) = -\frac{x}{(x+y)^2}\tiny.\end {array}\] We use this result to compute the partial derivatives of \(\frac{\partial}{\partial x}\!f(x,y)\) so as to find the second order partial derivatives of #f#: \[ \begin{array}{rcl}\frac{\partial}{\partial x}\!\!\left(\frac{\partial}{\partial x}\! f(x,y)\right) &=& \frac{\partial}{\partial x}\!\!\left(\frac{y}{(x+y)^2}\right) = -2\frac{y}{(x+y)^3} \\ \frac{\partial}{\partial y}\!\!\left(\frac{\partial}{\partial x}\! f(x,y)\right) &=& \frac{\partial}{\partial y}\!\!\left(\frac{y}{(x+y)^2}\right) = \frac{xy}{(x+y)^3}\\ \frac{\partial}{\partial x}\!\!\left(\frac{\partial}{\partial y}\! f(x,y)\right) &=& \frac{\partial}{\partial x}\!\!\left(-\frac{x}{(x+y)^2}\right) = \frac{xy}{(x+y)^3} \\ \frac{\partial}{\partial y}\!\!\left(\frac{\partial}{\partial y}\! f(x,y)\right) &=& \frac{\partial}{\partial y}\!\!\left( -\frac{x}{(x+y)^2}\right) = -2\frac{x}{(x+y)^3} \\\end {array}\] What is striking here is that the "mixed" derivatives \(\frac{\partial}{\partial x}\!\!\left(\frac{\partial}{\partial y}\! f(x,y)\right)\) and \(\frac{\partial}{\partial y}\!\!\left(\frac{\partial}{\partial x}\! f(x,y)\right)\) are equal to each other. This is no coincidence: see the following theorem.

Commutation of mixed derivatives If the partial derivatives of \(f(x,y)\) exist and are continuous, then the order of the partial derivatives in higher derivatives is irrelevant: \[\frac{\partial}{\partial y}\!\!\left(\frac{\partial}{\partial x}\! f(x,y)\right)=\frac{\partial}{\partial x}\!\!\left(\frac{\partial}{\partial y}\! f(x,y)\right)\]

Notation for higher derivatives The following notation is used for derivatives of second order: \[ \begin{array}{ccccccc} \frac{\partial}{\partial x}\!\!\left(\frac{\partial}{\partial x}\! f(x,y)\right) & = & \frac{\partial^2}{\partial x^2}\! f(x,y) & = & \frac{\partial^2f}{\partial x^2}\! (x,y) & = & f_{xx}(x,y) \\ \\ \frac{\partial}{\partial x}\!\!\left(\frac{\partial}{\partial y}\! f(x,y)\right) & = & \frac{\partial^2}{\partial x\partial y}\! f(x,y) & = & \frac{\partial^2f}{\partial x\partial y}\! (x,y) & = & f_{xy}(x,y)\\ \\ \frac{\partial}{\partial y}\!\!\left(\frac{\partial}{\partial x}\! f(x,y)\right) & = & \frac{\partial^2}{\partial y\partial x}\! f(x,y) & = & \frac{\partial^2f}{\partial y\partial x}\! (x,y) & = & f_{yx}(x,y) \\ \\ \frac{\partial}{\partial y}\!\!\left(\frac{\partial}{\partial y}\! f(x,y)\right) & = & \frac{\partial^2}{\partial y^2}\! f(x,y) & = & \frac{\partial^2f}{\partial y^2}\! (x,y) & = & f_{yy}(x,y) \end {array}\] Similar notation is used for higher partial derivatives and partial derivatives of functions of more than two variables.

*Note:* in most textbooks the notation \(f_{yx}\) for \(\frac{\partial^2f}{\partial x\partial y}\! (x,y)\) used. Since, for functions with continuous derivatives, we have \(f_{xy}=f_{yx}\), it does not matter.

Indeed, when performing partial differentiation of \(f\) with respect to \(y\) we consider \(x\) as a constant. Because the derivative of the exponential function is equal to the exponential function, we get

\[ \begin{array}{rcl}\frac{\partial^2}{\partial y^2} \e^{x+y}&=&

\frac{\partial^2}{\partial y^2} \left(\e^{x}\cdot \e^{y}\right)\\

&&\phantom{xxxuvw}\color{blue}{\e^{x+y}=\e^x\cdot\e^y}\\&=&

\e^x\cdot \frac{\partial^2}{\partial y^2}\e^y\\&&\phantom{xxxuvw}\color{blue}{\e^{x}\text{ viewed as a constant}}\\&=&\e^x\cdot \frac{\partial}{\partial y}\!\!\left(\frac{\partial}{\partial y} \e^y\right)\\&&\phantom{xxxuvw}\color{blue}{\text{definition }\frac{\partial^2}{\partial y^2} }\\&=&\e^x\cdot \frac{\partial}{\partial y}\e^y\\&&\phantom{xxxuvw}\color{blue}{\text{derivative of }\e^y}\\&=&\e^x\cdot \e^y\\&=&\e^{x+y}\end {array}\]

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