Let [math]f\colon \mathbb{R}^{2}\to\mathbb{R}[/math] given by

[math]f(x,y)=\frac{xy^3}{x^2+y^6}[/math] if [math](x,y)\ne (0,0)[/math]

and 0 if (x,y)=(0,0)

 

Question asks me to use the definition or partial derivatives to show that f_x and f_y both exist at (0,0). and at any other point (x,y)

 

i can do the first part, but for the second part i cannot simplify the expression for the limit.

Can't you use the binomial formula or something like that ?

If you can do it at (0,0), i think you are surely pretty close to showing it for every couple (x,y)

 

Mandrake

The expression for the limit does simplify: first simplify the expression for f(x+h,y) - f(x,y) by making it into a single fraction, then after that you should find that all the terms on the top of the fraction that do not have a h in them will cancel, and then the other h's will cancel with the one on the bottom.