- #1

bobby2k

- 127

- 2

**Is there a "Leibnitz theorem" for sums with variable limits?**

Wikipedia says that if we want to differentiate integrals where the variable is in the limit and in the integrand, we can use Leibnitz theorem:

But what if I need to integrate a function defined like this:[itex]\Sigma_{I(x)}[f(x,t)][/itex], Here I(x) just means that the values depend on x. Or even what if I simpler:

[itex]\Sigma^{b(x)}_{a(x)}[f(x,t)][/itex], where we just have that the start and end of the summation depend on x. Are there some conditions where if we want to calculate:

[itex]\frac{d}{dx}[\Sigma_{I(x)}[f(x,t)]][/itex], we can move move the derivative inside, and get some more terms. Or do we have to calculate the sum before differentiating?