(Sorry for all the questions on this, I am really struggling with this and really appreciate all help.)

Show that the function $\displaystyle f(z)=|z|^2$ is differentiable at the origin.

So if $\displaystyle z=x+iy$, then $\displaystyle |z|^2=x^2+y^2$.

To show that it's differentiable you have to show that the limit of the difference quotient exists at 0:

$\displaystyle \lim_{\Delta z\to 0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}$

$\displaystyle =\lim_{\Delta z\to0}\frac{f(0+\Delta z)-f(0)}{\Delta z}$

$\displaystyle =\lim_{\Delta z\to0}\frac{(\Delta x)^2+(\Delta y)^2-0}{\Delta x+i\Delta y}$

Is this the right way to start?