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:

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

Second Derivative

Let $f \left({x, y}\right)$ 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:

$(1): \quad \dfrac {\partial^2 f}{\partial x^2} = \dfrac {\partial}{\partial x} \left({\dfrac {\partial f}{\partial x}}\right)$

$(2): \quad \dfrac {\partial^2 f}{\partial y^2} = \dfrac {\partial}{\partial y} \left({\dfrac {\partial f}{\partial y}}\right)$

$(3): \quad \dfrac {\partial^2 f}{\partial x \partial y} = \dfrac {\partial}{\partial x} \left({\dfrac {\partial f}{\partial y}}\right)$

$(4): \quad \dfrac {\partial^2 f}{\partial y \partial x} = \dfrac {\partial}{\partial y} \left({\dfrac {\partial f}{\partial x}}\right)$

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

Also see

• Definition:Jacobian Matrix
• Definition:Gradient

• Results about partial differentiation can be found here.

Sources

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