# Definition:Partial Derivative

## 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:

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

### Vector Function

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 $a_i$.

## 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.