Definition:Partial Derivative/Higher Derivative

Definition

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 $f$:

$\dfrac {\partial^4 u} { \partial x \partial y \partial z^2} := \map {\dfrac \partial {\partial x} } {\dfrac {\partial^3 u} {\partial y \partial z^2} } =: \map {f_{3 3 2 1} } {x, y, z}$