Definition:Partial Derivative/Second Derivative

Definition

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:

\(\text {(1)}: \quad\)

\(\ds \dfrac {\partial^2 f} {\partial x^2}\)

\(=\)

\(\ds \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial x} }\)

\(\ds =: \map {f_{1 1} } {x, y}\)

\(\text {(2)}: \quad\)

\(\ds \dfrac {\partial^2 f} {\partial y^2}\)

\(=\)

\(\ds \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial y} }\)

\(\ds =: \map {f_{2 2} } {x, y}\)

\(\text {(3)}: \quad\)

\(\ds \quad \dfrac {\partial^2 f} {\partial x \partial y}\)

\(=\)

\(\ds \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial y} }\)

\(\ds =: \map {f_{2 1} } {x, y}\)

\(\text {(4)}: \quad\)

\(\ds \dfrac {\partial^2 f} {\partial y \partial x}\)

\(=\)

\(\ds \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial x} }\)

\(\ds =: \map {f_{1 2} } {x, y}\)

Examples

Example: $u + \ln u = x y$

Let $u + \ln u = x y$ be an implicit function.

Then:

$\dfrac {\partial^2 u} {\partial y \partial x} = \dfrac {\partial^2 u} {\partial x \partial y} = \dfrac u {u + 1} + \dfrac {x y u} {\paren {u + 1}^2}$

Also see

• Clairaut's Theorem

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: $1.3$ Higher Order Derivatives
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Partial Derivatives: $13.60, \ 13.61$