Definition:Partial Derivative/Higher Derivative/Third Derivative

Definition

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 $f$:

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

Examples

Example: $u = \map \ln {x^2 + y}$

Let $u = \map \ln {x^2 + y}$ be a real function of $2$ variables such that $x^2 + y \in \R_{>0}$.

Then:

$\dfrac {\partial^3 u} {\partial y^2 \partial x} = \dfrac {\partial^3 u} {\partial x \partial y^2} = \dfrac {\partial^3 u} {\partial x \partial y \partial x} = \dfrac {4 x} {\paren {x^2 + y}^3}$

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: $1.3$ Higher Order Derivatives