Question

Define the first partial derivatives of a function \(f\) of two variables \(x\) and \(v\)

Short answer

The first partial derivative of a function \(f(x,v)\) is a derivative in which only one variable is considered (either \(x\) or \(v\)), while the other one is treated as a constant. This leads to two partial derivatives: one with respect to \(x\) (\(\partial f / \partial x\) or \(f_x\)) and the other with respect to \(v\) (\(\partial f / \partial v\) or \(f_v\)).

Step-by-step solution

Understanding functions of multiple variables

A function of two variables, denoted as \(f(x, v)\), has two inputs: \(x\) and \(v\). It could be written in the form of \(f(x, v) = x^2 + v^2\) or any other form where both \(x\) and \(v\) are involved.

Defining Partial Derivatives

The partial derivative of a function of two variables with respect to one of them is calculated by differentiating with respect to that variable, treating the other variable as a constant. For example, the partial derivative of \(f(x,v)\) with respect to \(x\), denoted as \(\partial f / \partial x\) or \(f_x\), is found by differentiating \(f\) with \(v\) treated as a constant.

First Partial Derivative with respect to \(x\)

The first partial derivative of \(f(x,v)\) with respect to \(x\) is written as \(\partial f / \partial x\) or \(f_x\). It is computed by taking the derivative of the entire equation with respect to \(x\), while treating \(v\) as a constant.

First Partial Derivative with respect to \(v\)

Similarly, the first partial derivative of \(f(x,v)\) with respect to \(v\) is written as \(\partial f / \partial v\) or \(f_v\). It is computed by taking the derivative of the entire equation with respect to \(v\), while treating \(x\) as a constant.