Solved: Consider the function defined by (a) Find and for (b) Use the definition of

Chapter 13, Problem 133

(choose chapter or problem)

Consider the function defined by

\(f(x, y)=\left\{\begin{array}{ll}

\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\

0, & (x, y)=(0,0)

\end{array}\right\).

(a) Find \(f_{x}(x, y)\) and \(f_{y}(x, y)\) for \((x, y) \neq(0,0)\).

(b) Use the definition of partial derivatives to find \(f_{x}(0,0)\) and \(f_{y}(0,0)\).

\(\left[\text { Hint: } f_{x}(0,0)=\lim _{\Delta x \rightarrow 0} \frac{f(\Delta x, 0)-f(0,0)}{\Delta x}\right]\)

(c) Use the definition of partial derivatives to find \(f_{x y}(0,0)\) and \(f_{y x}(0,0)\).

(d) Using Theorem 13.3 and the result of part (c), what can be said about \(f_{x y}\) and \(f_{y x}\).

Text Transcription:

f(x,y)=fracxy(x^2-y^2t)x^2+y^2,&(x,y)neq(0,0)0,&(x,y)=(0,0)

f_y(x,y)

f_x(x,y)

(x, y)neq(0,0)

f_x(0,0)

f_y(0,0)

[Hint:f_x(0,0)=lim_Deltax0fracf(Deltax,0)-f(0,0)Deltax]

f_xy(0,0)

f_xy

f_yx