Definition:Partial Derivative

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Definition

Real Analysis

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:

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



Vector Function

Definition

Let $\map {\R^n} {x_1, x_2, \ldots, x_n}$ denote the real Cartesian space of $n$ dimensions.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.


Let $\mathbf f: \R^n \to \R^n$ be a vector-valued function on $\R^n$:

$\forall \mathbf x \in \R^n: \map {\mathbf f} {\mathbf x} := \ds \sum_{k \mathop = 1}^n \map {f_k} {\mathbf x} \mathbf e_k$

where each of $f_k: \R^n \to \R$ are real-valued functions on $\R^n$.


For all $k$, let $f_k$ be differentiable at $a$.


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

$\map {\dfrac {\partial \mathbf f} {\partial x_i} } {\mathbf a} := \ds \sum_{k \mathop = 1}^n \map {g_{k i} '} {a_i} \mathbf e_k$

where:

$g_{k i}$ is the real function defined as $\map {g_k} {x_i} = \map {f_k} {a_1, \ldots, x_i, \dots, a_n}$
$\map {g_{k i}'} {a_i}$ is the derivative of $g_k$ at $a_i$.


Cartesian $3$-space

Let $\map {\R^3} {x, y, z}$ denote the Cartesian $3$-space.

Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.

Let $\mathbf V$ be a vector field in $\R^3$.


Let $\mathbf v: \R^3 \to \mathbf V$ be a vector-valued function on $\R^3$:

$\forall P = \tuple {x, y, z} \in \R^3: \map {\mathbf v} P := \map {v_1} P \mathbf i + \map {v_2} P \mathbf j + \map {v_3} P \mathbf k$


Let $v_1, v_2, v_3$ be differentiable at $\mathbf a = \tuple {a_x, a_y, a_z}$.


The partial derivatives of $\mathbf v$ with respect to $x$, $y$ and $z$ at $\mathbf a$ are denoted and defined as:

$\map {\dfrac {\partial \mathbf v} {\partial x} } {\mathbf a} := \map {\dfrac {\d v_1} {\d x} } {x, a_y, a_z} \mathbf i + \map {\dfrac {\d v_2} {\d x} } {x, a_y, a_z} \mathbf j + \map {\dfrac {\d v_3} {\d x} } {x, a_y, a_z} \mathbf k$
$\map {\dfrac {\partial \mathbf v} {\partial y} } {\mathbf a} := \map {\dfrac {\d v_1} {\d y} } {a_x, y, a_z} \mathbf i + \map {\dfrac {\d v_2} {\d y} } {a_x, y, a_z} \mathbf j + \map {\dfrac {\d v_3} {\d y} } {a_x, y, a_z} \mathbf k$
$\map {\dfrac {\partial \mathbf v} {\partial z} } {\mathbf a} := \map {\dfrac {\d v_1} {\d z} } {a_x, y, a_z} \mathbf i + \map {\dfrac {\d v_2} {\d z} } {a_x, a_y, z} \mathbf j + \map {\dfrac {\d v_3} {\d z} } {a_x, a_y, z} \mathbf k$

Complex Analysis

Definition:Partial Derivative/Complex Analysis

Value at a Point

Let $\map f {x_1, x_2, \ldots, x_n}$ be a real function of $n$ variables

Let $f_i = \dfrac {\partial f} {\partial x_i}$ be the partial derivative of $f$ with respect to $x_i$.


Then the value of $f_i$ at $x = \tuple {a_1, a_2, \ldots, a_n}$ can be denoted:

$\valueat {\dfrac {\partial f} {\partial x_i} } {x_1 \mathop = a_1, x_2 \mathop = a_2, \mathop \ldots, x_n \mathop = a_n}$

or:

$\valueat {\dfrac {\partial f} {\partial x_i} } {a_1, a_2, \mathop \ldots, a_n}$

or:

$\map {f_i} {a_1, a_2, \mathop \ldots, a_n}$

and so on.


Hence we can express:

$\map {f_i} {a_1, a_2, \mathop \ldots, a_n} = \valueat {\dfrac \partial {\partial x_i} \map f {a_1, a_2, \mathop \ldots, a_{i - 1}, x_i, a_{i + i}, \mathop \ldots, a_n} } {x_i \mathop = a_i}$

according to what may be needed as appropriate.


Second Derivative

Let $\map f {x, y}$ be a function of the two independent variables $x$ and $y$.


The second partial derivatives of $f$ with respect to $x$ and $y$ are defined and denoted by:

\(\text {(1)}: \quad\) \(\ds \dfrac {\partial^2 f} {\partial x^2}\) \(=\) \(\ds \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial x} }\) \(\ds =: \map {f_{1 1} } {x, y}\)
\(\text {(2)}: \quad\) \(\ds \dfrac {\partial^2 f} {\partial y^2}\) \(=\) \(\ds \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial y} }\) \(\ds =: \map {f_{2 2} } {x, y}\)
\(\text {(3)}: \quad\) \(\ds \quad \dfrac {\partial^2 f} {\partial x \partial y}\) \(=\) \(\ds \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial y} }\) \(\ds =: \map {f_{2 1} } {x, y}\)
\(\text {(4)}: \quad\) \(\ds \dfrac {\partial^2 f} {\partial y \partial x}\) \(=\) \(\ds \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial x} }\) \(\ds =: \map {f_{1 2} } {x, y}\)


Higher Derivative

Higher partial derivatives are defined using the same technique as second partial derivatives.

Examples are given for illustration:


Third Partial Derivative

Let $u = \map f {x, y, z}$ be a function of the $3$ independent variables $x$, $y$ and $z$.

The following is an example of one of the $3$rd derivatives of $f$:

$\dfrac {\partial^3 u} {\partial z^2 \partial y} := \map {\dfrac \partial {\partial z} } {\dfrac {\partial^2 u} {\partial z \partial y} } =: \map {f_{2 3 3} } {x, y, z}$


Fourth Partial Derivative

Let $u = \map f {x, y, z}$ be a function of the $3$ independent variables $x$, $y$ and $z$.

The following is an example of one of the $4$th derivatives of $f$:

$\dfrac {\partial^4 u} { \partial x \partial y \partial z^2} := \map {\dfrac \partial {\partial x} } {\dfrac {\partial^3 u} {\partial y \partial z^2} } =: \map {f_{3 3 2 1} } {x, y, z}$


Order

$u = \map f {x_1, x_2, \ldots, x_n}$ be a function of the $n$ independent variables $x_1, x_2, \ldots, x_n$.

The order of a partial derivative of $u$ is the number of times it has been (partially) differentiated by at least one of $x_1, x_2, \ldots, x_n$.

For example:

a second partial derivative of $u$ is of second order, or order $2$
a third partial derivative of $u$ is of third order, or order $3$

and so on.


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


Examples

Notation for 3-Value Function

Let $u = \map f {x, y, z}$ be a real function of $3$ variables.

Then the partial derivatives may be expressed variously as:

$\dfrac {\partial u} {\partial x} = \map {f_1} {x, y, z} = \dfrac {\partial f} {\partial x} = \map {\dfrac \partial {\partial x} f} {x, y, z}$
$\dfrac {\partial u} {\partial y} = \map {f_2} {x, y, z} = \dfrac {\partial f} {\partial y} = \map {\dfrac \partial {\partial y} f} {x, y, z}$
$\dfrac {\partial u} {\partial z} = \map {f_3} {x, y, z} = \dfrac {\partial f} {\partial z} = \map {\dfrac \partial {\partial z} f} {x, y, z}$


Arbitrary Cubic

Let $\map z {x, y}$ be the real function of $2$ variables defined as:

$z = x^3 - 3 x y + 2 y^2$

Then we have:

\(\ds \dfrac {\partial z} {\partial x}\) \(=\) \(\ds 3 x^2 - 3 y\)
\(\ds \dfrac {\partial z} {\partial y}\) \(=\) \(\ds -3 x + 4 y\)
\(\ds \dfrac {\partial^2 z} {\partial x^2}\) \(=\) \(\ds 6 x\)
\(\ds \dfrac {\partial^2 z} {\partial y^2}\) \(=\) \(\ds 4\)
\(\ds \dfrac {\partial^2 z} {\partial x \partial y} = \dfrac {\partial^2 z} {\partial y \partial x}\) \(=\) \(\ds -3\)


Example: $x z^y$

Let $\map f {x, y, z} = x z^y$ be a real function of $3$ variables.

Then the partial derivative with respect to the $2$nd variable may be expressed as:

$\map {f_2} {x, y, z} = x z^y \ln z$

and because of the notation chosen, we have:

$\map {f_2} {r, s, t} = r t^s \ln t$


Also see

  • Results about partial differentiation can be found here.


Sources



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