Partial Derivative/Examples/Arbitrary Cubic

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