Differentiability of the function $f(z)=|z|^2$.

Differentiability of the function $f(z)=|z|^2$.

I'm stuck on the following problem:
The function $f(z)=|z|^2$ is:
Differentiable only at the Origin
Not differentiable anywhere
I have to determine which of the aforementioned options is true. The
answer key to the problem says option 2 is true whereas I think option 1
is correct.
We see that: $$ f(z)=|z|^2 \implies
u(x,y)+iv(x,y)=x^2+y^2,\;\text{where}\:z=x+iy,\:\text{say}. $$ Then: $$
u(x,y)=x^2+y^2,v(x,y)=0.\;\text{So at the Origin}\;u_x=u_y=v_x=v_y=0 $$
So, C-R equation is satisfied, and hence option 1 holds true.
Am I going in the right direction?