Manipulating identities I'm having some trouble deriving certain identities. If
$$S(z) = \prod_{i=1}^n (z-z_i)$$
then how can I write 
$$\frac{1}{S(z)}\frac{d^2S}{dz^2} = \sum_{i=1}^n\frac{1}{z-z_i}\sum_{j\neq i}^n\frac{2}{z_i-z_j}$$
and
$$ \frac{1}{S(z)}\frac{dS}{dz}= \sum_{i=1}^n\frac{1}{z-z_i} $$
Truth be told, I'm having some trouble writing the derivative of $S(z)$ in a neat, compact form.

$\frac { dS } { dz} =\sum _ { i = 1 } ^ n\Pi_{j\neq i}(z-z_j)$
So $ \frac { 1 } {S ( z)}\frac { dS } { dz}=\sum _ { i = 1 } ^ n\Pi_{j\neq i}(z-z_j)/(\Pi_{i=1}^n(z-z_j))=\sum _ { i = 1 } ^ n\frac { 1 } {z - z_i}$
Now can you do the second derivative?
