Main thing: Chain rule
So as a 'modified' version of the fundamental theorem of calculus: Let . Then .
Imagine and .
Apply the fundamental theorem of calculus as you would normally to get:
Take the derivative of and you're done.
Find the derivative of
h(x) = integral(sin(x) at the top and -5 at the bottom) of
So... here's what I thought the answer should be, but I'm evidently missing something, again.
My plan of attack is to insert sin(x) for t and then subtract the function with -5 inserted for t:
cos(sin^5(x)) + sin(x) - (cos((-5)^5) - 5)
I'm getting told this is incorrect. Any glaring mistakes? I've been doing this for several hours, so I may have gone mentally blind.
Okay, I see what you mean, but in earlier problems I've been finding the answer by plugging in the upper limit and subtracting the lower limit. This had been giving me the correct answer until this question.
For example, and earlier question said to find the derivative of upper limit = 1 lower = x
sin(t^3) The correct answer ended up being sin(1^3) - sin(x^3) or just -sin(x)
If I had followed the FTC and just used the upper limit I would've just gotten sin(1^3) or 0. How do I know when to do which?