Determining domain for C^1 function

[lys04]

Homework Statement

Picture

Relevant Equations

Partial derivatives

The ##$x partial derivative is equal to L \frac{4x}{5(x^{2}+y^{2})^{\frac{-3}{5}}} and the partial for$y$is L \frac{4y}{5(x^{2}+y^{2})^{\frac{-3}{5}}} Using the limit definition of partial derivatives I got the partial wrt$x$is L \frac{h^{\frac{4}{5}}}{h} which doesn’t exist as$h$goes to$0$. Similar argument for partial wrt$y$. This means that$f$isn’t$C^1$at the origin, right? At every other point the partial derivatives exist and is continuous because it’s a composition of a polynomial of two variables and$x^2/5$, so$f$is$C^1## at all points except the origin.

Is the reasoning correct?

 

[lys04]

I don’t think latex is working, not sure what’s wrong with it sorry

 

[Hill]

I don’t think latex is working, not sure what’s wrong with it sorry

One problem is that { and } don't match.

 

[Orodruin]

Works very well with the additional }:

$$L \frac{4x}{5(x^{2}+y^{2})^{\frac{-3}{5}}}$$

 

[pasmith]

$\sqrt[2n+1]{x}$ is not differentiable at $x = 0$ for $n \geq 1$. This follows from the fact that $x^{2n+1}$ has a point of inflection here.