First derivative of a summation (looking for the steps)

Question

What is the first derivative $\dfrac{dF}{d\zeta}$ of the following function

F = $\zeta+\sum_{k=1}^{N} \dfrac{m_k}{\zeta^k}$

Thank you!

Answer 1

Well, we have:

$$\frac{\partial}{\partial\zeta}\left\{\zeta+\sum_{\text{k}=1}^\text{N}\frac{\text{m}_\text{k}}{\zeta^\text{k}}\right\}=\frac{\partial}{\partial\zeta}\left(\zeta\right)+\sum_{\text{k}=1}^\text{N}\text{m}_\text{k}\cdot\frac{\partial}{\partial\zeta}\left\{\frac{1}{\zeta^\text{k}}\right\}=1+\sum_{\text{k}=1}^\text{N}\text{m}_\text{k}\cdot\left(-\frac{\text{k}}{\zeta^{\text{k}+1}}\right)\tag1$$