# Thread: Check differentiability in function

1. ## Check differentiability in function

Hello, I can't find any way to prove if this funtion is or isn't differentiable in if $\displaystyle (x,y)=(0,0)$:

$\displaystyle {f(x,y)=\displaystyle\frac{x^{3}}{x^{2}+y^{2}}}$ if $\displaystyle (x,y) \neq(0,0)$

$\displaystyle f(x,y)=0$ if $\displaystyle (x,y)=(0,0)$

Thank you!

2. Originally Posted by Bop
Hello, I can't find any way to prove if this funtion is or isn't differentiable in if $\displaystyle (x,y)=(0,0)$:

$\displaystyle {f(x,y)=\displaystyle\frac{x^{3}}{x^{2}+y^{2}}}$ if $\displaystyle (x,y) \neq(0,0)$

$\displaystyle f(x,y)=0$ if $\displaystyle (x,y)=(0,0)$

Thank you!
Start by taking the derivative and seeing whether or not it is defined at $\displaystyle (0,0)$.

3. Originally Posted by Bop
Hello, I can't find any way to prove if this funtion is or isn't differentiable in if $\displaystyle (x,y)=(0,0)$:

$\displaystyle {f(x,y)=\displaystyle\frac{x^{3}}{x^{2}+y^{2}}}$ if $\displaystyle (x,y) \neq(0,0)$

$\displaystyle f(x,y)=0$ if $\displaystyle (x,y)=(0,0)$

Thank you!
Use the following formulas to find $\displaystyle f_x(0,0)$ and $\displaystyle f_y(0,0)$:
$\displaystyle f_x(a,b)=\lim_{h\to0} \frac{f(a+h,b)-f(a,b)}{h}$
$\displaystyle f_y(a,b)=\lim_{h\to0} \frac{f(a,b+h)-f(a,b)}{h}$

I have calculated derivative respect x: $\displaystyle \dfrac{\partial f}{\partial x}=\displaystyle\frac{x^{4}+3x^{4}y^{2}}{{(x^{2}+y ^{2})}^{2}}$, respect y: $\displaystyle \dfrac{\partial f}{\partial y}=\displaystyle\frac{-2yx^{3}}{{(x^{2}+y^{2})}^{2}}$ but then?
5. I have calculate that limit of $\displaystyle \dfrac{\partial f}{\partial x}$ doesn't exist.. How can I continue?