general form of coefficients in partial fraction decomposition Given 
$$ \frac{P(s)}{Q(s)} = \frac{A_{1}}{s-r_{1}} + \dots + \frac{A_n }{s-r_n},$$
where $P(s)$ is a polynomial with degree less than $n$ and $Q(s)$ is a polynomial with degree $n$ and with $r_1 ,\dots , r_n$ distinct zeros. 
I'm trying to show that 
$$ A_k = \frac{P(r_k)}{Q'(r_k)} , \quad k = 1 , \dots , n,$$
with which I'm having trouble.
My attempt is as follows , 
\begin{gather}
\frac{P(s)}{Q(s)} = \frac{A_1 \sum_{j \neq 1}^{n}(s-r_j) + \dots + A_n \sum_{j \neq n}^{n}(s-r_j)}{\prod\limits_{j\le n} (s-r_j)} = \frac{\sum_{i=1}^{n} A_i \left(\sum_{j\neq i}^{n} (s-r_j)\right)}{\prod\limits_{j\le n} (s-r_j)} \\
\frac{P(s)}{Q'(s)} = \frac{\sum_{i=1}^{n} A_i \left(\sum_{j\neq i}^{n} (s-r_j)\right)}{\sum_{j=1}^{n+1} (s-r_j)' \prod\limits_{1 \le i \le n+1 \\ \ \ \ \ \ j \neq i}(s-r_i)}
\end{gather}
Honestly I'm not even sure what I'm doing at this point, I don't know if I'm taking the right avenue, there must be another elegant way. Any tips and hints would be appreciated.

Multiply both sides by $s-r_k$.
Then it follows that $$A_k=\lim_{s\to r_k}(s-r_k)\frac{P(s)}{Q(s)}=\frac{P(r_k)}{Q'(r_k)}.$$
Can you fill in the details?
