Definition:Partial Derivative/Second Derivative

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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:

\(\text {(1)}: \quad\) \(\displaystyle \dfrac {\partial^2 f} {\partial x^2}\) \(=\) \(\displaystyle \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial x} }\) \(\displaystyle =: \map {f_{1 1} } {x, y}\)
\(\text {(2)}: \quad\) \(\displaystyle \dfrac {\partial^2 f} {\partial y^2}\) \(=\) \(\displaystyle \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial y} }\) \(\displaystyle =: \map {f_{2 2} } {x, y}\)
\(\text {(3)}: \quad\) \(\displaystyle \quad \dfrac {\partial^2 f} {\partial x \partial y}\) \(=\) \(\displaystyle \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial y} }\) \(\displaystyle =: \map {f_{2 1} } {x, y}\)
\(\text {(4)}: \quad\) \(\displaystyle \dfrac {\partial^2 f} {\partial y \partial x}\) \(=\) \(\displaystyle \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial x} }\) \(\displaystyle =: \map {f_{1 2} } {x, y}\)


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


Sources