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.): $\S 13.3$