How to prove differentiable everywhere?

[athrun200]

Homework Statement

See photo, part b and c

Homework Equations

The Attempt at a Solution

For part b
It seems it is trival, in part a we have proved that $f_{x}$ and $f_{y}$ exist. Obviously, they are differentiable for x and y$\neq$0

For part c.
It seems there are 2 method to do it.
1. Use first principle.(i.e. take limit)
2.Find $f_{xy}$ and $f_{yx}$
If they equal each other, then f is differentiable at(0,0)

 

[HallsofIvy]

Homework Statement

See photo, part b and c

Homework Equations

The Attempt at a Solution

For part b
It seems it is trival, in part a we have proved that $f_{x}$ and $f_{y}$ exist. Obviously, they are differentiable for x and y$\neq$0

For part c.
It seems there are 2 method to do it.
1. Use first principle.(i.e. take limit)

What limit are you talking about? "Differentiability" of a function of two variables is NOT defined by a limit. You might want to look up the definition of "differentiable" for functions of more than one variable.

2.Find $f_{xy}$ and $f_{yx}$
If they equal each other, then f is differentiable at(0,0)

No, that's not true. There is a theorem that says "If a function is differentiable, on a region, then its mixed second derivatives are equal on that region", but the converse of that statement is not true.

 

[athrun200]

How about part b?
You only talk about part c

 

[HallsofIvy]

Yes, (b) is trivial- f is a quotient of two polynomials and the denominator is not 0 for any point other than (0, 0).

 

[athrun200]

What limit are you talking about? "Differentiability" of a function of two variables is NOT defined by a limit. You might want to look up the definition of "differentiable" for functions of more than one variable.

But I saw from wiki that it is.
If not, how do I prove differentiable?

 

[athrun200]

I also wonder if $f_{x}$=$\frac{(2^xy^3-6x^2y^2-2x^4)}{(y^4+2x^2y^2+x^4)}$ exist at (0,0).

If not, it seems the question part a has some problems.