I need to figure out how $\sum_{k=1}^{n} \text{ln}\ \lambda_{k} = - \frac{d}{ds}\Big|_{s=0} \sum_{k=1}^{n} \lambda_{k}^{-s}$.
But, if I evaluate $- \frac{d}{ds}\Big|_{s=0} \sum_{k=1}^{n} \lambda_{k}^{-s}$, I get $- \sum_{k=1}^{n} \Big( \frac{d}{ds} \lambda_{k}^{-s} \Big) \Big|_{s=0}$, which becomes $ \sum_{k=1}^{n} \Big( s\ \lambda_{k}^{-s-1} \Big) \Big|_{s=0}$, and this is $0$.
Where am I making the mistake?
Hint:
$$ \lambda^{-s}_k=e^{-\log(\lambda_k)s} $$
I'm sure you can take it from here
essentialy you are differentiating w.r.t. to $\lambda$ which is not what you want!