How do I take the derivative of an integral in this question?

Question

rogue · Answer

Fundamental theorem of calculus?
What's your question?

anonymous · Answer

attachment

anonymous · Answer

Yes, it's based on ftc

anonymous · Answer

all of them

anonymous · Answer

Yep, all of them :)

rogue · Answer

$$\int\limits_{0}^{x}f(t)dt = f(x)$$Here's an example on solving FTC problems.$$\int\limits_{x^2}^{x^3} t^2 dt = (x^3)^2 \frac {d}{dx} x^3 - (x^2)^2 \frac {d}{dx} x^2$$

anonymous · Answer

How did you guys determine that?

rogue · Answer

You substitute the limits of integration, and multiply it by the derivative of the limit of integration.

anonymous · Answer

ohh...I understand how you did that.  Thanks!

anonymous · Answer

@Rogue think simple!
Derivative cancel out the integration!

rogue · Answer

It is simple, but you have 2 variables =P

anonymous · Answer

In here, just 1 variable x :)

rogue · Answer

The x substitutes into the t.

anonymous · Answer

Rogue, your example comes out to x^5 3x^2 - x^4 2x.  How do you subtract that to get the answer?

rogue · Answer

You just leave it, you can't subtract. 
I was showing you stuff with x^2 and x^3 because often times students only substitute the limit of integration into the variable, without multiplying by the derivative of the limit of integration.

anonymous · Answer

oh, okay.  Thanks again!

anonymous · Answer

www.youtube.com/watch?v=PGmVvIglZx8
is definitely worth a look

anonymous · Answer

@Rogue, you proves noteworthy point because it did happen right in OS!

rogue · Answer

:)

What's OS?

rogue · Answer

Oh, openstudy, lol.