Firstly, isn't the rule that subtraction of powers is division not multiplication?
Then on finding the integral
why has the e to 3 come out as negative?
It is all the same. Please ponder it until you see that it is not scary.
Feel free simply to remove constants from the integral argument. is a constant.
By the way, "the derivative of e" is zero. Be more careful with words.
perhaps it will make more sense if you realize that .
this is a consequence of the chain rule: is really the composition:
, where g(x)= -x = (-1)x.
so when we integrate, we need a minus sign to go inside the integral, and another minus sign to go outside (to balance, (-1)(-1) = 1). the minus sign inside the integral disappears, since when we integrate:
the minus sign outside the integral stays, however.
Got it. Great explanations.