Definition:Partial Derivative/Real Analysis

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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:

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



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:

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

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


Sources