1.17. Let \( f: \mathbb{R}^{2} \rightarrow \mathbb{R} \) 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
\(\frac{\partial f}{\partial y}(0,0) = \lim_{k \to 0} \frac{f(0, k) - f(0, 0)}{k} = 0 \)
Step 1 ：\(\frac{\partial f}{\partial x} = \frac{4x^3}{2\sqrt{x^4 + y^4}} = \frac{2x^3}{\sqrt{x^4 + y^4}} \)
Step 2 ：\(\frac{\partial f}{\partial y} = \frac{4y^3}{2\sqrt{x^4 + y^4}} = \frac{2y^3}{\sqrt{x^4 + y^4}} \)
Step 3 ：\(\frac{\partial f}{\partial x}(0,0) = \lim_{h \to 0} \frac{f(h, 0) - f(0, 0)}{h} = 0 \)
Step 4 ：\(\frac{\partial f}{\partial y}(0,0) = \lim_{k \to 0} \frac{f(0, k) - f(0, 0)}{k} = 0 \)
