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:
\(\ds \dfrac {\partial z} {\partial x}\) | \(=\) | \(\ds 3 x^2 - 3 y\) | ||||||||||||
\(\ds \dfrac {\partial z} {\partial y}\) | \(=\) | \(\ds -3 x + 4 y\) | ||||||||||||
\(\ds \dfrac {\partial^2 z} {\partial x^2}\) | \(=\) | \(\ds 6 x\) | ||||||||||||
\(\ds \dfrac {\partial^2 z} {\partial y^2}\) | \(=\) | \(\ds 4\) | ||||||||||||
\(\ds \dfrac {\partial^2 z} {\partial x \partial y} = \dfrac {\partial^2 z} {\partial y \partial x}\) | \(=\) | \(\ds -3\) |
Proof
All results follow from Power Rule for Derivatives and the definition of partial derivative.
$\blacksquare$
Sources
- 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation