[-/1 Points]DETAILSSCALCET9 3.2.008.Differentiate g(x) (*+9 xJe*Need Help?Rand te[0/1 Points]DETAILSPREVIOUS ANSWERSSCALCET9 3.XP.2DifferentiateNeed Help?RedliVorin...

Differentiate each function. $$g(x)=\left(x^{3}-8\right) \cdot \frac{x^{2}+1}{x^{2}-1}$$

Okay, We're gonna find the derivative of this function. So first we have a polynomial three x squared, so we can just do six x to the one power. OK, minus two comes along. And now we just have to take the derivative of the coastline. And the derivative of the coastline is negative. The sign of X. So the answer six X plus to sign it. Kicks Uh huh, Yes.

Okay, so we're gonna be using the product rule to find the derivative function G. Prime of X. And so the product rule tells us if we have two functions F. X and G of X, and they're being multiplied. In this case F of X will be equal to nine X to the eighth minus eight X to the ninth, and G of X will be equal to X plus one over X. Then the derivative of that function is equal to the derivative of F. Of X. So the derivative of nine X to the eighth -8 X to the 9th, multiplied by G. Fx which was X plus one divided by X. And then we have plus F. Of X, which was just nine X to the eighth minus eight X to the ninth. And I'm running out of room here a little bit so I'm gonna make this a little bit smaller really quickly. And then so this last term is multiplied by the derivative of G. Of X. So the derivative of X plus one divided by X. And now that we have our derivative like this we need to find the derivative of nine X. To the eighth minus eight X. To the ninth and X plus one over X. So let's go ahead and find those two derivatives. We can say the derivative of nine next to the eighth minus eight, X to the ninth is equal to the derivative of nine next to the eighth minus the derivative of Fedex to the ninth. And now we can take both of these constants out of the derivatives. So we're gonna have nine times X to the eighth. Are the drift of XD 8 -8 times the derivative of X to the 9th. And now these two derivatives are just power rules. And so we can find these two derivatives pretty easily. The first one is going to be 72 times x to the seventh, and the second one is going to be 72 times X to the eighth. We brought down the power, We multiplied it by whatever constants we had and then we minus one to the power. So now we found this first derivative. Let's go ahead and find the derivative of X Plus one divided by X. So that's equal to the derivative of X plus the derivative of one divided by X. And so this is equal to the derivative of X is equal to one. And then we have plus, I'm just going to change this to the derivative of X to the negative first power. And here we can use the power rule um to find this derivative. And so this would be equal to one minus X. Two negative second power I brought down the power and then I minus one to the power. So we have negative one times X. To the negative one minus one power. Or negative X two negative second power. And so now that we have these two derivative functions, let's go ahead. And first I'm just going to move this stuff. Mhm. First thing I need to get it all. There we go. So now that I've moved down here left myself a little bit of room to actually write this out. And so the derivative was 72 X. To the seventh minus 72 X. To the eighth, multiplied by X plus one divided by X. And then we have plus and I'm just gonna put this term below. So nine X to the 8th minus eight X. To the ninth. And then multiplied by one minus one divided by X. Square. And this is our derivative function G. Prime of X.

So this is our function in European. Basically, apply the product rule and the change you'll simultaneously. Product rule is simply when your flu functions and their product, and we need to calculate a derivative calculate its derivative simply equal to view. Dash me plus Vida shoe on the changeable states that if you have the composition of flu functions, so if you have a four G affects, we need to calculate its decorator with its equal to the derivative off the outer function into the derivative of the inner function Fate. So considering three X minus one to the par seven as you on two. X plus one to the part five as R. V. And applying the product on the change would simultaneously we get this is equal toe. So you dash three X minus one to the par six now into the derivative off three X minus one, which is G affects in our case or inter function simply into three. So you dash. This is are you dash into me so two x plus one to the par five plus we'd ash becomes five into two x plus one to the car four into two, which is again the composition in a function G off access to X plus one. So G dash of access simply toe Andi, all of this multiplied by you, which is simply three x minus one to the par 70. So this is that, um, basically derivative. And we conduce all off, simplified on Sort of condense it. How much I will we like. So this becomes 21 into Yeah. Yeah. Oh, okay. And now, if we won't weaken, take three x minus on and do X plus one, also common. This simply becomes this entire scene multiplied by. So we're left with one toe ax plus one over here. Andi, we're left with 13 X minus 70 hell, T X minus one. Sorry and left until

The given function is G X. He goes to seven divided by four multiplied by X squared minus T v X left. To work in this caution, we need to find a derivative of Cuban function to apply the differentiation on this function with respect to X so it can return is Judas X equals two mhm be by DX seven divided by four went to carried by X squared minus three x left Well, so it becomes day by day x 70 Wanted by food X squared Do you by DX to the X plus du by DX right, Since it is a constant, um so it can be written as day by day X x squared minus three x It is a consent into it will be zero. So the differentiation of this time is equal to seven, divided by four multiplied by two x minus three So it becomes seven. Divided by two x minus city. It is the derivative of the given function