Problem 8 (!! 7 pts). Let $f(x, y) = \begin{cases} \frac{xy}{\sqrt{x^2 + y^2}} & (x, y) \neq (0, 0), \\ 0 & (x, y) = (0, 0). \end{cases}$ a) Compute $\frac{\partial f}{\partial x}(0, 0)$ and $\frac{\partial f}{\partial y}(0, 0)$. b) Using the definition, show that $f$ is not differentiable at $(0, 0)$.
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The partial derivative with respect to x is given by: ∂f/∂x = lim(h→0) [f(0+h,0) - f(0,0)]/h Since f(x,y) = x^2 + y^2, we can substitute these values into the equation: ∂f/∂x = lim(h→0) [(0+h)^2 + 0^2 - 0^2 - 0^2]/h = lim(h→0) [h^2]/h = lim(h→0) Show more…
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