# Definition:Partial Derivative/Real Analysis

## Definition

### At a Point

Let $U \subset \R^n$ be an open set.

Let $f: U \to \R$ be a real-valued function.

Let $a = \tuple {a_1, a_2, \ldots, a_n}^\intercal \in U$.

Let $f$ be differentiable at $a$.

Let $i \in \set {1, 2, \ldots, n}$.

#### Definition 1

The partial derivative of $f$ with respect to $x_i$ at $a$ is denoted and defined as:

$\map {\dfrac {\partial f} {\partial x_i} } a := \map {g_i'} {a_i}$

where:

$g_i$ is the real function defined as $\map g {x_i} = \map f {a_1, \ldots, x_i, \dots, a_n}$
$\map {g_i'} {a_i}$ is the derivative of $g$ at $a_i$.

#### Definition 2

The $i$th partial derivative of $f$ at $a$ is the limit:

$\map {\dfrac {\partial f} {\partial x_i} } a = \ds \lim_{x_i \mathop \to a_i} \frac {\map f {a_1, a_2, \ldots, x_i, \ldots, a_n} - \map f a} {x_i - a}$

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 $x$.

### On an Open Set

Let $U\subset\R^n$ be an open set.

Let $f : U \to \R$ be a real-valued function.

Let $f$ be differentiable in $U$.

The $i$th partial derivative (function) of $f$ with respect to $x_i$ is the real-valued function which sends each $x\in U$ to the $i$th partial derivative at $x$.

## Notation

There are various notations for the $i$th partial derivative of a function:

• $\dfrac {\partial f} {\partial x_i}$
• $\dfrac {\partial} {\partial x_i} f$
• $\map {f_{x_i} } {\mathbf x}$
• $\map {f_{x_i} } {x_1, x_2, \cdots, x_n}$
• $f_{x_i}$
• $\partial_{x_i}f$
• $\partial_i f$
• $D_i f$
• $\dfrac {\partial z} {\partial x_i}$
• $z_{x_i}$

where $z = \map f {x_1, x_2, \cdots, x_n}$.