1. ## Partial derivative question

Let f(x,y) = $\frac{x^{2}(x - y)}{x^{2} + y{2}}$ if (x,y) doesnt equal (0,0) and

0 if x = y = 0

a) Find $f_x$(0,0) and $f_y$(0,0)

b) Is f continuous at (0,0)?

c) Is f differentiable at (0,0)? Explain.

Attempt:

a) $F_x$(0,0) and $f_y$(0,0) both = 0

b) No since f(x,y) = 0 (if x = y = 0) and $\frac{x^{2}(x - y)}{x^{2} + y{2}}$ (if (x,y) doesnt equal (0,0)

c) It isnt differentiable at (0,0) similar reasoning as b

2. a)

$\displaystyle f_{x} (0,0) = \lim_{\delta \rightarrow 0} \frac{1}{\delta}\ \frac{\delta^{3}}{\delta^{2}} = 1$

$\displaystyle f_{y} (0,0) = \lim_{\delta \rightarrow 0} \frac{1}{\delta}\ (-\frac{0}{\delta^{2}}) = 0$

Kind regards

$\chi$ $\sigma$

3. Originally Posted by SyNtHeSiS
Let f(x,y) = $\frac{x^{2}(x - y)}{x^{2} + y{2}}$ if (x,y) doesnt equal (0,0) and

0 if x = y = 0

a) Find $f_x$(0,0) and $f_y$(0,0)

b) Is f continuous at (0,0)?

c) Is f differentiable at (0,0)? Explain.

Attempt:

a) $F_x$(0,0) and $f_y$(0,0) both = 0
It would be better if you showed how you got these- then we could comment. $f_x$ is certainly NOT 0. Since f(0, 0)= 0, the partial derivative with respect to x is give by
$\lim_{h\to 0}\frac{f(h, 0)- 0}{h}= \lim_{h\to 0}\frac{\frac{h^2(h)}{h^2+ 0}}{h}= \lim_{h\to 0}\frac{h^3}{h^3}$

b) No since f(x,y) = 0 (if x = y = 0) and $\frac{x^{2}(x - y)}{x^{2} + y{2}}$ (if (x,y) doesnt equal (0,0)
What? You appear to be saying that any non-constant function is not continuous!
What is the limit of f as (x, y) goes to (0, 0)?

c) It isnt differentiable at (0,0) similar reasoning as b
What reasoning? All you did was repeat the definition of the function!
what is the definition of "differentiable" for a function of two variables?