Indices, Combining Sum and Product Notation

Question

I have a function $$ f(x) = (x-a_1)(x-a_2) \cdot \cdot \cdot (x-a_n) $$

I want to express $f'$ with sigma notation. It is clear that $f'$ is the sum of $n$ terms, where each term is missing one $(x-a_i)$. So I want to sum over n terms, and for each term I want to multiply all but one $(x-a)$ in the original polynomial.

I think it is: $$ f'(x) = \sum_{\substack{i = 1}}^n \prod_{\substack{j \neq i}}^n \;(x-a_j) $$ An alternative is $$ f'(x) = \sum_{i = 1}^n \prod_{\substack{i = 1 \\j \neq i}}^n \;(x-a_j) $$

Does this say what I want it to say?

Edit:

The correct notation is $$ f'(x) = \sum_{\substack{i = 1}}^n \prod_{\substack{j = 1 \\ j \neq i}}^n \;(x-a_j) $$

Answer 1

Yes it says what you want it to say. For every $i,$ you're only multiplying the $(x-a_j)$'s that are not equal to $(x-a_i).$

Answer 2

An elegant alternative is to index over a set. For $j \in \{1,\dots,n\}$, let $I_j := \{1,\dots,n\} \backslash\{j\}$.

Now, if $$f(x) = \prod_{j=1}^n (x-a_j),$$ then $$f'(x) = \sum_{j=1}^n \prod_{k \in I_j}(x-a_k).$$