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Nead Help?Points]DETAILSSCALCET8 2.8.027.Find the derivative of the function using the definition of derivative. g(x) V 39'(x)State the domain of the function ...

Question

Nead Help?Points]DETAILSSCALCET8 2.8.027.Find the derivative of the function using the definition of derivative. g(x) V 39'(x)State the domain of the function (Enter your answer using intervalState the domain of its derivative. (Enter your answer using intervalNeed Help?ReadtWatch ItTalkito TutorPFFVIOUS HNSWERSISCALCET

Nead Help? Points] DETAILS SCALCET8 2.8.027. Find the derivative of the function using the definition of derivative. g(x) V 3 9'(x) State the domain of the function (Enter your answer using interval State the domain of its derivative. (Enter your answer using interval Need Help? Readt Watch It Talkito Tutor PFFVIOUS HNSWERSI SCALCET



Answers

Find the derivative of the function. Find the domains
of the function and its derivative.
$g(x)=\cos ^{-1}(3-2 x)$

Ffx is equal to extra power. Three over to the first thing about me so determined off dysfunction is all riel numbers. So the venous from negative infinity to positive infinity. Now, the next thing we need to do is we need to find the derivative. So FX, by definition, is living. Each tends to zero f affects plus H minus F x over each so that people to limit each tends to zero X plus a h to the power 3/2 minus X to the power 3/2 over h. So what we're gonna do here is gonna rewrite this, Likely different. So we're gonna say this is limit H chance to zero X plus a h to the power 1/2 raised to the power three and minus X power 1/2 raised to the power three over each. So this now is off. The form is squared minus bl in Q minus. Be cute. So going back to our basic factoring thank you minus B cube can be factored as he minus b times his way plus a B plus B squared. So what that means is, if I say is equal to x plus age to the power 1/2 and I say B is equal to extremely power 1/2. Then I can factor this X plus h to the power 1/2 raised to the power three minus x Probable 1/2 race to be poverty can be fact it as I was an A minus bi so that would be tex plus H of power 1/2 minus X power 1/2 and then that would be times is quit so it's quit will be X plus a H and then plus a B. So that would be plus X plus a h to the power 1/2 times X to the power 1/2 and then plus B squared. So that would be plus X. So this means that now we can replace our this year with that expression. So we have limit. H tends to zero, and this would be it's plus H power 1/2 minus X power one f and then times So x plus each plus X plus a h power 1/2 squared times X power 1/2 and then plus X or H so we can further simplify this just a little bit So we say limit age 10 steals your and then X plus a h power 1/2 minus X power 1/2 And then times this becomes two X plus h And then times, uh, plus x plus a h power 1/2 kind X polar. What? Half over h. So far, so good. Now we're gonna use something called the community property of multiplication, which simply says that two factors are being multiplied. Then you can switch their places. So I'm gonna write this as two x plus age plus X plus h power 1/2 times its power. What half? And then times expressed each power 1/2 minus x power of what happens. All I've done here is interchange fee positions off those two factors. Now, the next thing we're gonna do is we're gonna take care of this second factor in the new miniter. And how we do that is using your technique corn rationalizations. So let me show you how it works. So first of all, that you just copy the whole thing as it is so hot. X plus age plus explosives for over half times. It's our 1/2 and then times it's plus th power 1/2 minus X power 1/2 and then divided by each. So we're gonna multiply this with a fraction. And the fracture that people to multiply is what? Because the conjugated off this second factor here. So this will be X plus age power 1/2 plus x power 1/2 and then in community after I would have expressed h power 1/2 plus x polar 1/2. Now why am I doing that? So this is off before he minus b times a plus B. So I'm just gonna write this out here. So a minus b times a plus B and that is equal to a squared minus piece quick. So if you consider express h square root as a and then consider extra de Pollo 1/2 SB then express age problem 1/2 minus X power 1/2 times expressive power one have plus Explorer 1/2 that would become a squared minus B squared. So yummy. Right? Get out here. So we have limit. Each tends to zero. So we got two x plus a H plus x plus h power 1/2 times x power 1/2 and then the other factor. This would be square off express, each under the square root, so that would be just express age and then minus X over h times X plus a h power of 1/2 times X power 1/2. So what we see happening there is a way we haven't x, that we have a minus X so they will cross out and then we would have limit. Each tends to zero two x plus each plus X plus age power one halftime X power 1/2 times each or each times X plus a h power 1/2 plus x power 1/2 and not the agent The denominator and age in the new major they will cancel each other out and now we are at the point where we can actually applied the limit. So now we can substitute each age as your and that would give you two x plus X power of 1/2 times X power 1/2 or e x power 1/2. Thus X power 1/2 so that then simplifies to two x plus x over to X power 1/2 so that here's me in the new later three X over to power X 1/2 and that simplifies to 3/2 X power 1/2. So there is the derivative off my function. And now Sparty domain for this function goes the domain for this function would be X is greater than is you. Because we cannot have negative numbers in the uh huh squared. So this would be a meanness from zero to infinity.

So in this problem were given this function F of X is X to the To the 3/2. And were asked to do several things. Were asked to find the domain of this, the derivative of using the function and the domain of the derivative. So, I want to start with the domain here, first of all. So the domain is 02 infinity. As I cannot do a negative number To the 1/2 power. Okay. So there's my domain for my function. Now, definition of derivative definition derivative says F prime of X is equal to the limit as H goes to zero of F of X plus H minus F of X over H. All right. So for us then that means derogative fx is the limit as H goes to zero of X plus H two, the three house minus X to the three house all over H. And we can see real quick is that if we multiply this by X plus H two the three halves plus X to the three house over X plus H three halves plus X to the three halves. Right? Because what are we doing? We're doing a let's see, we're doing a minus B times A plus B equals a squared minus b squared. All right. So that means this is the limit. His page goes to zero of x plus age to the three halves squared. Well, the one half power squared cancels out. And so I'm left with X plus H cubed minus X cubed, aren't I? Because extra three halves squared would be Next to the 3rd. Okay. Age times X plus H. 23 halves plus X. To the three halves. All right now multiply. This out Limit is H goes to zero. Uh X cubed plus three X square at H plus three X. H squared plus H cubed minus X cubed over. H times X plus H. Two. The three halves Plus X. 2 3/2. All right. And what do we notice? First of all we noticed that X cubed minus X cubed. So that's gone next. We noticed that I got an H. Here in the denominator and H. And that term and H there and I can take one of those ages cancel the H. Is out. Okay And so I'm left with what I'm left with the limit as H goes to zero of three x squared Plus three XH plus H squared over X plus H. Two the three halves plus X to the three halves. Okay now perform the limit Well when h goes to zero that term goes to zero that term goes to zero and that H goes to zero doesn't it? Some left with three X squared over execute was X cubed. Extra three halves plus extra three halves. That's two X. to the three. Well I didn't want to write very well let's try that again. X. to the three house. So that's three over to well X squared over extra three halves. That's X. To the two minus three halves in the export up there. So that leaves me X to the one half, doesn't it? Okay. And the domain then is still zero to infinity. Just like we said above. Right, We can't do a negative number to the one half, unless we can't do the square root of a negative number. So here is my derivative and its domain.

For the income, we want to find the derivative of the function using the function or using the definition of the derivative. So we know the definition of the derivative. Is this right here? The function that we're given is three plus X Over 1 -3 x. Still plugging in exports H here and expose each year. Then we subtracted by after backs And divide that by age, letting the limit go to zero. When we simplify though, we'll end up getting 10 over 1 -3 x. Where that is gonna end up being the derivative. NBC that the domains are going to be all real numbers without Um 1/3. If x equals 1/3, then this would be undefined.

So we have. Ffx is in. Quote two X squared minus two. Exe cute so that the main of dysfunction is all of the numbers. Domain is from negative. Infinity too positive. Infinity. Let's find out the derivative off dysfunctions O F Prime off excess limit H tends to zero F off X plus H minus F X over each, so that would be limit. Each tends to zero X plus each squared minus X plus a h cute minus X squared minus two extra deep our three, So there should be a A to Indian as well. So then divide by each, so this would be limit. Age tends to zero X squared plus two X age plus each square minus two X cubed, plus the X squared egx plus three x each squared minus H cute and then minus X squid plus two x cute over h. So before we proceed, we can simplify things a little bit in the minute it is, and we can eliminate this. So then going forward, this will really make hte tends to zero two x h plus States Quay minus two x cute, minus six X squared, each minus six x each squared and minus two h cute and then plus two x cute over each. So can we can eliminate the minus two x cubed and plus two x cute. And then we can start dividing his turn by HR We could actually say that H is factored up so limited. Each tends to his units factor each other. So we have two x plus each minus six X square minus six x each and minus two H square over each. And then we can eliminate this X here and this each year. So the to each one of the numerator one in the dominator will cross out And now what we can do is we can actually substitute eight is equal to zero. And when we do that, we get two x minus six x squared. So the derivative off in function so f prime of X is equal to two x minus six x squared and the domain of the function is the domain of derivative is also from negative infinity to positive. Infinity


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