Integral and differential of summation

[Jhenrique]

The following identities are true?
$$\frac{d}{dx} \sum_{u_0}^{u_1}f(x,u)\Delta u = \sum_{u_0}^{u_1}\frac{d}{dx}f(x,u)\Delta u$$ $$\int \sum_{u_0}^{u_1}f(x,u)\Delta u dx = \sum_{u_0}^{u_1}\int f(x,u)dx\Delta u$$

 

[HallsofIvy]

As long as $(u_1- u_0)/\Delta x$ is finite, those can be proved by induction.

 

[Jhenrique]

As long as $(u_1- u_0)/\Delta x$ is finite, those can be proved by induction.

I didn't understand...