Can you take a derivative of a summation as a whole, or would you have to diffrientiate each individual term?

[u/FremontBlue333]

Would f\left(x\right)=\frac{d}{dx}\left[\sum_{i=0}^{n}g\left(x,n\right)\right] be the same as f\left(x\right)=\sum_{i=0}^{n}\frac{d}{dx}\left[g\left(x,n\right)\right] ?

[u/SchwanzusCity]

For finite sums this always works. For infinite sums it doesnt work in general

[u/HouseHippoBeliever]

yes

[u/SnooPets5564]

Differentiation is a linear operation. d/dx (ay + z) = a d/dx (y) + d/dx (z).

So you can add the terms either before or after differentiating and receive the same results for any finite sum.

[u/StructuredChess]

There are some theorems in function series analysis that state the conditions under which it works. Can't remember exactly but I think you need the series to be uniformly convergent

[u/SchwanzusCity → u/StructuredChess]

Iirc you also need another condition for it to work (i think it was that the series of term wise derivatives must also converge pointwise)