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

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

Mohamed Khalil
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    As written, this expression doesn't make much sense. I assume you mean for $n$ to range over all natural numbers $n$ such that $1\leq n\leq f(x)$; but, if that is the case, the sum only changes when $f(x)$ crosses integer boundaries. – Nick Peterson Jan 29 '19 at 19:40

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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.

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?
Joe
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  • Not true. For instance, if $f(x)=1$, then $F$ is constant and is $\mathcal C^\infty$. Another example, if $f(x)=1+\cos(x)$, then $F$ is piece-wise constant, and can be differentiated everywhere, except at discrete points. In fact, it is differentiable in the sense of distributions, with the derivative being a sum of Dirac distributions. – Stefan Lafon Jan 29 '19 at 20:00
  • It is often possible to extend a function, defined in terms of a summation, so that it is well defined even when the upper limit of the summation is not an integer. – R. Burton Jan 29 '19 at 20:29