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Recently I had to perform the following differentiation for some index $i\in[0,M] \cap \mathbb{N}$ $$\frac{d}{dw_i}\left\{\sum_{j=0}^M w_j^2x^{2j} +2 w_jx^j\sum_{k

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    $i\in[0,M]$ or $ i\in[0,M]\cap \mathbb{N}$? – 2017-02-02
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    @MathOverview $ i \in [0,M] \cap \mathbb{N}$, i is some index. – 2017-02-02

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Note that $$ \left( w_j^2x^{2j} + 2 w_jx^j\sum_{ki \end{array} \right. $$ implies $$ \frac{d}{d w_i} \left( w_j^2x^{2j} + 2 w_jx^j\sum_{ki \end{array} \right. $$ Then calculate the derivative $\frac{d}{d w_i}$ of \begin{array}{rl} \displaystyle\sum_{\substack{0\leq j\leq M}} \left( w_j^2x^{2j} + 2 w_jx^j\displaystyle\sum_{k