Partial Derivative/Examples/Arbitrary Cubic

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Examples of Partial Derivatives

Let $\map z {x, y}$ be the real function of $2$ variables defined as:

$z = x^3 - 3 x y + 2 y^2$

Then we have:

\(\ds \dfrac {\partial z} {\partial x}\) \(=\) \(\ds 3 x^2 - 3 y\)
\(\ds \dfrac {\partial z} {\partial y}\) \(=\) \(\ds -3 x + 4 y\)
\(\ds \dfrac {\partial^2 z} {\partial x^2}\) \(=\) \(\ds 6 x\)
\(\ds \dfrac {\partial^2 z} {\partial y^2}\) \(=\) \(\ds 4\)
\(\ds \dfrac {\partial^2 z} {\partial x \partial y} = \dfrac {\partial^2 z} {\partial y \partial x}\) \(=\) \(\ds -3\)


All results follow from Power Rule for Derivatives and the definition of partial derivative.