# 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:

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

### 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

- 1961: David V. Widder:
*Advanced Calculus*(2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: $1.1$ Partial Derivatives

- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards:
*Calculus*(8th ed.): $\S 13.3$