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Compute partial derivatives of the given function

by RoRi

Let

    \[ f(x,y) = x^4 + y^4 - 4x^2 y^2. \]

Compute all first-order and second-order partial derivatives. Verify that the mixed partial derivatives are equal.

The first-order partial derivatives are given by

    \begin{align*} \frac{\partial f}{\partial x} &= 4x^3 - 8xy^2; \\ \frac{\partial f}{\partial y} &= 4y^3 - 8x^2 y. \end{align*}

The second partial derivatives are given by

    \begin{align*} \frac{\partial^2 f}{\partial x^2} &= \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x} \right) \\ &= \frac{\partial}{\partial x} (4x^3 - 8xy^2) \\ &= 12x^2 - 8y^2. \\ \frac{\partial^2 f}{\partial y \partial x} &= \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) \\ &= \frac{\partial}{\partial y} (4x^3 - 8xy^2) \\ &= -16xy.\\ \frac{\partial^2 f}{\partial x \partial y} &= \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) \\ &= \frac{\partial}{\partial x} (4y^3 - 8x^2y) \\ &= -16xy.\\ \frac{\partial^2 f}{\partial y^2} &= \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial y} \right) \\ &= \frac{\partial}{\partial y} (4y^3 - 8x^2y) \\ &= 12y^2 - 8x^2. \end{align*}

From this we see that the mixed partials are equal:

    \[ \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} = -16xy. \]

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