Math Problem Statement
Solution
The problem you shared asks to use the definition of partial derivatives as limits to find the partial derivatives and for the given function , which is defined piecewise:
\begin{cases} \frac{-xy(x^2 - y^2)}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0) \end{cases}$$ ### Step 1: Definition of partial derivatives The partial derivative of $$f$$ with respect to $$x$$ at $$(0, 0)$$ is given by the limit definition: $$f_x(0, 0) = \lim_{h \to 0} \frac{f(h, 0) - f(0, 0)}{h}$$ Similarly, for the partial derivative with respect to $$y$$: $$f_y(0, 0) = \lim_{h \to 0} \frac{f(0, h) - f(0, 0)}{h}$$ ### Step 2: Computing $$f_x(0, 0)$$ To compute \( f_x(0, 0) \Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Partial Derivatives
Limits
Formulas
f_x(0, 0) = lim(h -> 0) [f(h, 0) - f(0, 0)] / h
f_y(0, 0) = lim(h -> 0) [f(0, h) - f(0, 0)] / h
Theorems
Definition of Partial Derivatives
Suitable Grade Level
College Level (Calculus III or Advanced Calculus)
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