Find the derivative of $\displaystyle f(z)=5z^2-10z+8$ using the following definition:

$\displaystyle f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}$

I started with

$\displaystyle f'(z_0)=\lim_{z\to z_0}\frac{5z^2-10z+8-(5z_0^2-10z_0+8)}{z-z_0}$

which simplifies a bit to

$\displaystyle f'(z_0)=\lim_{z\to z_0}\frac{5z^2-10z-5z_0^2+10z_0}{z-z_0}$

but where do you go from there? Thanks!