Definition:Partial Derivative

Definition
Let $f\left({x_1, \ldots, x_n}\right)$ be a real-valued function of multiple independent variables.

The partial derivative of $f$ 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.

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 $x$".

Function on Real Vector Space
Let $\left({x_1, x_2, \ldots, x_n}\right)$ be considered as a vector in the vector space $\R^n$.

Then the partial derivative of $f$ can be defined as follows:

Also denoted as
Other notations are:


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


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

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

Also see

 * Definition:Gradient