derivative w.r.t summation limits

Question

I wonder if someone can help me to find the differentiation with respect to the upper limit of a summation as shown below !!

$\frac{d}{dx}\sum\limits_{n=1}^{f(x)} g(n)$.

Answer 1

I'm assuming you mean the upper limit is the least integer less than or equal to $f(x)$. Since $g(n)$ has no dependence on $x$, the function $$F(x) = \sum_{n = 1}^{f(x)}g(n)$$ is not continuous, let alone differentiable.

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The comments point out two important things:

1. This analysis has some edge cases where it isn't true. For exmaple, if $1 \leq f(x) < 2$ or if $g(n) = 0$ for all $n > 1$, then this will be a constant function. Then it is continuous and differentiable.
2. There are ways to extend sums to non-integer indices. It didn't seem to me like this is what was asked for, but if there is interest, I direct you to this post. What is the derivative of a summation with respect to its upper limit?