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CM2 Prac Question 7.5

Laura

Very Active Member
Hi all,
Would you be able to further explain how to obtain the differentiation of the RHS of the equation?

I'm unsure why the summation term for i for both terms disappears.

Thanks!
 
In a more general case let's consider:
\[I = \sum_{i=1}^{n}\sum_{j=1}^{n}x_i x_j\]
When taking partial derivatives with respect to \(x_k\) we assume that all other \(x\)'s remain constant. So the only terms that will have a non-zero value are those which involve \(x_k\) specifically, ie:
\begin{align}
\frac{\partial I}{\partial x_k} & = \sum_{i=1}^{n}\sum_{j=1}^{n}\frac{\partial x_i x_j}{\partial x_k}\ \\
& = \underbrace{\sum_{i=1}^{n}x_i}_{\rm{when}\: j=k} + \underbrace{\sum_{j=1}^{n}x_j}_{\rm{when}\: i=k} \\
& = 2\sum_{i=1}^{n}x_i \\
\end{align}
Please let me know if that's not what you're asking about.
 
Last edited:
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