Multivariate Functions

A multivariate function is a function with several variables or argument. For example, \(f(x)=x^2\) is a uni-variate function with only one single variable \(x\), while the function \(f(x_1,x_2)=x_1^2+3x_2\) is a multivariate function with two variables \(x_1\) and \(x_2\). The multivariate functions can be written in a general form as \(f(x_1,x_2,\dots,x_n)\)  with  \(n\) variables \(x_1,x_2,\dots,x_n\). If we put all the \(x_i\) variables in a single vector \(x\) as

\[x=\begin{bmatrix} x_1\\x_2\\ \vdots\\x_n \end{bmatrix}\]

the multivariate function \(f(x_1,x_2,\dots,x_n)\)  can be also written as \(f(x)\). 

Partial Derivatives

A partial derivative of a function of several variables is its derivative with respect to one of the variables, assuming the other variables are constant. If a multivariate  function \(f(x_1,x_2,\dots,x_n)\), the partial derivative of \(f\) with respect to each \(x_i\) is denoted by

\[\partial f/\partial x_i \quad \text{or} \quad \partial_{x_i } f\]

For example, let's assume that \(f(x_1,x_2)=x_1^2+3x_2\). Then the partial derivatives with respect to \(x_1\) and \(x_2\) are given as: \[\partial f/\partial x_1=2x_1, ~ ~ ~ ~ \partial f/\partial x_2=3.\]