Definition:Partial Derivative/Value at Point

Definition

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.

Sources

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