Derivative of complex function using definition

**The question**

Show that the definition of $\displaystyle \frac{d}{dz}(\frac{1}{1 - z}) = \frac{1}{(1 - z)^2}$

**My attempt**

I tried using:

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

But having fractions in the numerator in this case appears difficult to simplify. I tried substituting z = x + iy, but I ended up with a massive mess which still results in a division by 0. Is there a good method of solving this problem?

Thanks.