Someone asked this question: and I am very interested in the answer, but don't understand it and don't have enough reputation to comment on it directly.
The question asks if you can solve something like: $$ g(x) = \int_a^b f(x) \,dx$$ by differentiating both sides, and then saying $$g'(x) = f(x)$$.
The answerer says that you can ignore the $a$ and $b$ boundaries on the definite integral. This is really convenient, but I don't understand why. $\int_a^b f(x) \,dx$ is surely different than $\int_a^c f(x) \,dx$, where $b \neq c$.
Can anyone help me understand the intuition behind this?