Chapter 6.2, Problem 69E

(a)

To determine

To calculate: The values of $f_x\left(0,y\right)$ by calculating the limit $\underset{h\to 0}{\lim }\left(\frac{f\left(h,y\right)-f\left(0,y\right)}{h}\right)$ where function f is defined as $f\left(x,y\right)=\{\begin{array}{l}\frac{xy\left(x^2-y^2\right)}{x^2+y^2},\text{for }\left(x,y\right)\ne \left(0,0\right)\\ 0\text{ , for }\left(x,y\right)=\left(0,0\right)\end{array}$.

(b)

To determine

To calculate: The values of $f_y\left(x,0\right)$ by calculating the limit $\underset{h\to 0}{\lim }\left(\frac{f\left(x,h\right)-f\left(x,0\right)}{h}\right)$ where function f is defined as $f\left(x,y\right)=\{\begin{array}{l}\frac{xy\left(x^2-y^2\right)}{x^2+y^2},\text{for }\left(x,y\right)\ne \left(0,0\right)\\ 0\text{ , for }\left(x,y\right)=\left(0,0\right)\end{array}$.

(c)

To determine

To calculate: The values of the function $f_{yx}\left(0,0\right)$ and $f_{xy}\left(0,0\right)$ where function f is defined as $f\left(x,y\right)=\{\begin{array}{l}\frac{xy\left(x^2-y^2\right)}{x^2+y^2},\text{for }\left(x,y\right)\ne \left(0,0\right)\\ 0\text{ , for }\left(x,y\right)=\left(0,0\right)\end{array}$ and also compare $f_{yx}\left(0,0\right)$ and $f_{xy}\left(0,0\right)$ for given function.