Definition:Partial Derivative

Definition
Given a function of multiple independent variables $f \left({x_1, \ldots, x_n}\right)$, the partial derivative with respect to $x_i$ is denoted and defined as:


 * $\dfrac{\partial f}{\partial x_i} = \dfrac{\mathrm d g}{\mathrm d x_i}$

where:
 * $g \left({x_i}\right) = f \left({x_1, \ldots, x_i, \dots, x_n}\right)$
 * $\dfrac{\mathrm d g}{\mathrm d x_i}$ is the derivative of $g$ with respect to $x_i$
 * all the $x_j, j \ne i$ are considered as constant.

If $\left({x_1,x_2,\cdots,x_n}\right)$ is considered as a vector in the vector space $\R^n$, then the partial derivative can be defined as follows.

Let:


 * $f: \R^n \to \R, \mathbf x \mapsto f\left({\mathbf x}\right)$

be a real-valued function where:


 * $\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$

is a vector in $\R^n$.

Let $\mathbf x^T$ denote the transpose of $\mathbf x$.

The partial derivative of $f$ with respect to $x_i$ is defined as:

provided such a limit exists.

Other notations are:


 * $\dfrac {\partial z}{\partial x_i}$
 * $\dfrac {\partial f}{\partial x_i}$


 * $f_{x_i} \left({x_1,x_2,\cdots,x_n}\right)$
 * $f_{x_i}$
 * $z_{x_i}$

where $z = f\left({x_1,x_2,\cdots,x_n}\right)$.

When spoken, $\dfrac {\partial y}{\partial x}$, "the partial derivative of $y$ with respect to $x$" is often shortened to "partial $y$ partial $x$", or "del $y$ del $x$".

Also see

 * Gradient