Complex analysis using definition of the derivative 
Question: 
  $f(z) = z + 2iz^2 \operatorname{Im}(z)$
Is the function differentiable at $z = 0$?
Where is $f(z)$ analytic?

Is there any way to do this using the definition of a derivative? 
I tried plugging in $z=x+iy$ and I got to the point $-2x^2y-4xi(y^3)+x-iy+2y^3$ and now I'm stumped. 
There is a way using Cauchy but I'm not too sure. Can anyone give me a head start?

I am not sure, because of your notation, which  is exactly the function that you mean,
 but  as a rule of thumb: functions that depend on $\overline{z}=Re(z)-Im(z)$ are not derivable. As your function depends in a single place of  on $Im(z)=\frac{z-\overline{z}}{2}$, your function have a dependence  on $\overline{z}$ and therefore it is not derivable.  
