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B_ a. by by Find dx 0.4.55 the I 41 8 derivative the the 12t dt I and directly: differentiating the result....

Question

B_ a. by by Find dx 0.4.55 the I 41 8 derivative the the 12t dt I and directly: differentiating the result.

b_ a. by by Find dx 0.4.55 the I 41 8 derivative the the 12t dt I and directly: differentiating the result.



Answers

Find the derivatives. a. by evaluating the integral and differentiating the result. b. by differentiating the integral directly. $$\frac{d}{d x} \int_{0}^{x^{3}} e^{-t} d t$$

If this problem were given this folded expression in port eight year as the first take integral. So integrate and then take the air. It is our friendship. All right, let's start with doctors. We have in jungle zero to square up, exposing tbt. We know that the entire effort about 70 scientist e So we will have the ddx of science t you wouldn't have zero. It's Greg affects. All right, this is equal to DDX of sign spirit of X minus Sign zero, which size you're zero. So that is a veritable science period of X. There were 12. Sign is, um, consigned spirit of eggs. Now there were dropped in a function which is spirit of eggs is one over to a spirit of X. So for for a refund this in part be here asked you differentiate the integral. So basically, we're as to use equation to in your textbooks. So we know that dd X So they've given injury go the same t t t and speak of two, um, cool side. Oh, this guy screwed up eggs times the derivative Oh, that was furtive. Ends the rebound on this to be percent spirit of X times more or to discredit X and a party. As you see, we find those

So fundamental Theorem of calculus here basically says that Ah integral sze and inverse of derivatives when you're looking at a derivative right next to an integral So the animals a year and 1/2 it's Cho san t d t One way we're gonna do this, which is what they say in part B differentiating the integral directly for part B. We're just gonna go ahead and cancel those out. But for part A, it has to scoot around on the hard way. So for part A, we're gonna do the evaluate the integral and then differentiate the results. So do you by the ex is gonna hang out and then I'm gonna do integral co sign. So the integral co sign a sign Evaluate that from zero to square root of ax. So I'm gonna take the derivative, huh? That would be so sign of spirit of X minus effort, a sign of zero sign a 00 So I'm gonna do the derivative of ah, the derivative of sign of X. And so the derivative sign of X is co sign with spirit of X. But then we need to multiply by the derivative because of the chain are also times 1/2 next to the negative 1/2 now in part B, when we have the by the X zero the square root of ax co Santy d T. You can think about canceling the integral with the derivative, but there's one PC Still need to remember. It's that square root of X. So all we're gonna do is plug in square, Rita asks, and then chain rule it.

A number by moving the formula 18. You say that the over knee next integration from a regio rex. Oh, Dean, you continue was equal to air. Yes, GXE times g dash Well, that's we really get the over gxe the integration from me who executed one over tgt She is equal to one over. Execute one over. Execute times X yield Ah dash which is equal to one over x q times three s square which is equal to three over its So we will achieve this results using the over Vieques configuration from 12 x cubes, one over tea the 80 she is equal to ah, then teen from 14 x cubed J is equal to Lin X cubed minus then one and then one is equal to zero. So people to land execute So it well, we get the derivative he over the x or men executes is equal to one over x a few times Ah, time three x squared which is equal to three overs Do you think went to its former? Ah, trying to So the question Bearden the same Oppose ah the or the x off the integration from one Britain s when people were teen 18. You able to leap over Lin x Times? You didn't x the relative just equal to X time Going over active equal to one and win with check. Ah, my making the, um the integration. We're just for me. She's e ah team integration from one to the next which is equal to you. Berlin X minus a poor one. Ah, which is equal to X minus e. If we take the derivative for this formula, it will be one. So this is the same value. So we check the

So in this problem, we are asked to find the durable knot This derivative, you see, of two differently. The first would be by evaluating this ventricle and then differentiating what we get and second iss by differentiating the integral direct. So we're gonna start with first week. So, uh, first we need to evaluate this empty growing We're going to preach or to integrate this. This is a power tool which mean Off T, which means we are going to use the powerful. So for the parable, when are integrating power? Um, ex, We, uh, want to add a one to the power. And then and then did I buy this new power? You said that's how we're going to do here the first step. Remember that we are not integrating, and we're not integrating it, and we're not driving it. So we need Teo. I'm sorry. We are integrating. We're not driving yet, so we need to keep this knot. Yeah. Okay, so here we don't put this in and we are going to ride. Yeah, Better wait till we have this. Three stays here. This's a coefficient. And now we want at one to the power and divide by this new power stand. Since this is a definite integral, we need to keep these limits. So first team C four to one until signing checks. Right. So, uh, we have here three and three, which must cancel out. And now we're ready, too. Evaluate from people to want to sign a tick sign. So next to we're still not driving. So we keep this the X, uh, when we start up with the top one. So we have sign cute effects, plus one time plus one. Cute. That is just one. Great. So now we can do it, right? So, for so we're going to direct by a turn by turn on this first term, we have function within a functions we have. We can think this's X cube with a sign of excess, a serum you end, which is essentially what we did here. So tonight, let's remember, General sir, we have x of G FX, and this is how we traps up in our kid's x f x is X cube. And inside the argument of F, we have sign it. All right. And since we need to know who Fujii Prime miss gonna figure that out right now. I thought we already did. Cho Cho san. Because this is how Where are you? All right, so now we're gonna use this Gino too. Careful. Derive this part set. Since we're driving, we no longer need does you can go ahead and race in so and prime and for directing f er the way we normally what? Treating kids X s the constant Wait. Bring down the street and we subtract one from it. So we bring down sign It's this new X and the exponents we subtract So we have three times sine square x and then we want to fly by Geo g Prime. That's co sign. Thanks. All right. Plus the derivative of a constant that it's syrup. So that means we don't need to write that. All right, so this is what this evaluates to. All right, So this is our answer, You see? First part. All right, so now we're going to go to the second second part. We're going different. She differentiate the integral directly. So to do that, we're going to use this, um, formula. We have the derivative with respect to X at an integral from a constant to a function of X off a function T and this girl respected P. Okay, we have more variable cell. Well, like maybe so. Hey, we have good news is we have this formula which will tell us exactly what this during this. Right. So So we're going to compare it to this part. They're going to compare this part too. Our original problem and figure out who FFT is and who you have. X iss then once we have that way, can figure out what this derivative is equal to. All right, so comparing we have there's hope t this will be a fft and this will be you have X. Our duty is to dream Teo Square and you effects ISS sign. Uh, thanks. All right, So, uh, so we have you vets and we have fft We want to figure out to a few exits, so we'll do that right now. All right, so now we need to plug in your ex in the argument of tp silly half, three times instead of t the right sign of X squared sign, son, if x squared, I ready. So, um, so again, using this formula, this is original problem so we can write and this other. All right, so we start with today counts sign. It's player A T. That is, Uh thanks, Terry. This's a Cuba X. And now we blow the prime five derivative with respect back a few of X. That is science. Thanks. Very So next we just want to simplify cuss because we know that this is co sign effects. So we just write that in tow This part wikileaks us this We don't need to change anything about it. So here we have using this other farms. You think that's from you? We have evaluated this integral doing. We have direct this interval drinks and this begins no.


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