1.17. Let $f : R^{2} \to R$ be the function:
$f (x, y) = \sqrt{x^{4} + y^{4}}$
(a) Find formulas for the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at all points $(x, y)$ other than the origin.
(b) Find the values of the partial derivatives $\frac{\partial f}{\partial x} (0, 0)$ and $\frac{\partial f}{\partial y} (0, 0)$ . (Note that the formulas you found in part (a) probably do not apply at $(0, 0)$ .

Answer

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Answer

$\frac{\partial f}{\partial y} (0, 0) = lim_{k \to 0} \frac{f (0, k) - f (0, 0)}{k} = 0$

Steps

Step 1 ： $\frac{\partial f}{\partial x} = \frac{4 x^{3}}{2 \sqrt{x^{4} + y^{4}}} = \frac{2 x^{3}}{\sqrt{x^{4} + y^{4}}}$

Step 2 ： $\frac{\partial f}{\partial y} = \frac{4 y^{3}}{2 \sqrt{x^{4} + y^{4}}} = \frac{2 y^{3}}{\sqrt{x^{4} + y^{4}}}$

Step 3 ： $\frac{\partial f}{\partial x} (0, 0) = lim_{h \to 0} \frac{f (h, 0) - f (0, 0)}{h} = 0$

Step 4 ： $\frac{\partial f}{\partial y} (0, 0) = lim_{k \to 0} \frac{f (0, k) - f (0, 0)}{k} = 0$