Diferentiate the function using one or more of the differentiation rules: y=(4x 5)7...

$7-46$ Find the derivative of the function
$$
U(y)=\left(\frac{y^{4}+1}{y^{2}+1}\right)^{5}
$$

In this case, we have to find a derivative of this function. So, uh, usedto you're gonna use the change rule. In this case, the deliberate about five resistible minus one x will be five days to the palm I minus one next time, 75 and then we differentiate the power, which is minus one over X. So in this case, the differentiation will be five days to the ball, minus one over x Times L and five and hear the difference station minus can be taken outside. It is a constant so and this can be written as deal with the X f x rays to the poor minus one. So this comes out as minus off five days to the poor, minus one over x times 75 and hear the difference station will be, uh, minus one times X rays to the ball minus two. So finally the answer comes sort of these two minus becomes plus, So we have five days to the palm, minus one over x times, Ellen five over X squared. So this is the value of the that

So the problem that we are solving today is finding the derivative of X plus one over X to the power of five. Right, So you can see that there is a power, and then there is an inside function. So that means that with the use of change room, um, so when you apply the power rule because that's the outside function you get five times when I was inside, you subtract to the X 1 to 4. So that's the outside function. Done the inside function. Um, distributive X is just one, because it's a linear function and the coefficient of X s one. And then to make it easier to, um, derive one of the X going to rewrite it as x, the power of negative one. So I'm going to play a power rule. All right. So that means I would bring the negative down and then keep thanks. And you subtract the exponents, you get negative, too. All right. And then So we want to rewrite the function so that the negative exponent is in the denominator. All right, so this will stay the same times one. Now it's going to be a minus because you distribute the negative, right? And then one over X squared, because to make a negative exponent positive, you bring it to the denominator. Mhm. So your final answer would be five times x plus one over x, their power four times one minus one over X squared.

Question 26. We need to find a derivative of this function. So since this is the ratio of the two functions, we gotta use the, uh, caution rule over here. So for the quotient rule, we have denominator square. So we have I square Plus to raise to the par five whole square way have denominator differentiation off numerator minus new marital differentiation off denominator, which is wise for a plus two y race to the pole five. So this consult us vice purpose to various to the 45 remains asters here. We have to use the change rules. So four comes down. Why, minus firepower decreases by once. That becomes three. And the transition of Y minus one is just one on for the next time we have y minus one. Raised to the 44 Here again, we gotta use while welcomes down. We have to use the chain rule. Why square plus two y power decreases by one and then we differentiate and say term, which is why square plus two y and the denominator remains as it is. So we here we have ice weapons to where is to the par 55 times to extend. So here we have 10. All right, so now let's simplify this further. We have y squared plus two y raised to the power five, and he will have four times why minus one whole cube. Minus five times Y minus one. Raised to about four times. Why square plus two y reached to about four on in order to finish it. This we have this two y plus two over wise weapons to wear these to the power 10. All right, so let's take, uh, we can take a few times out so we can take why Square pless do I raised to the ball? Five out along with y minus. I first not fire. We can take four out. So here we have. Full then. Here we have y minus one. Hold Cuba. So, what are we left with us? Four times. Why? Square plus tau minus five times. Why? Minus one times to wipe us too. Over. Why Square plus two y raised to the power. All right, so this is powerful. This is Port 10. So four power. So you get canceled and we're left with six hour off white scrappers to win. Denominator and the numerous, uh, Y minus One whole cube remains others. It's open of the brackets here, so we have four white square plus eight yr minus five times. This will be to White Square on duh. This is just to women's to wear zero. And this is minus two from one point of the brackets again. So we have four minus 10 four, minus 10 is minus six. So we have minus six wives world than we have eight y as it is. And here we'll have five times to us 10 or what? Why square plus two way race to the policies. So this is the required derivative.

So we have our function G of Y. And we're looking to find the derivative. So we want to find G prime of why? Um So to do that, there's as always there's many ways that we can approach this problem. So I think the best way we want to try to rewrite g f Y. So I'm going to get rid of this outside exponents here just to um just to get rid of some of the complexity of the problem. So what do we do? We just multiply the exponents. Right, so on top we have Y to the 10th, and then on bottom we have the quantity Y plus one Raised to the 5th power. Um So from here we could easily go ahead and use the quotient rule or we could move this up top and could use the product rule. So I always got and just use the quotient rule for this one. Um Since I've been using the product rule quite a lot, let's remind ourselves what the quotient rule is. So if we have two functions that are being divided, say F divided by G and we want to find the derivative, we take um G times F prime minus F G. Prime. And then on the bottom we take G and we square it. So just like with the product rule, we need to find the derivatives of the top and bottom first or the F and G rather. So in our case, F is wide at the 10th and G is Y plus one to the fifth. So first let's off to the side, let's write down all of these components to our formula is going to be, So f crime is going to be the derivative of Y to the 10th, Which is equal to 10 wide of the 9th so far. So good. And know what about G. Prime? So that's going to be the derivative of Y plus one to the fifth. So there's a little bit more complicated. So five comes outfront. Right? So we have five times Y plus one to the fourth and we need to take the derivative of the inside but derivative of white. This one is just one. So the general doesn't really do much there. So it's just gonna be five. Why? Plus 1 to the 4th. And our formula also requires G squared. So let's just go and write out what that would be down here as well, G squared would be Why? Plus 1 to the 5th? All that squared. So we're just gonna multiply these exponents. Uh So we would have Y plus one to the 10th. So now we've got all the components we need, we can go and use our product role to find our derivative G. Prime of Y. So we have G times F. Prime. So G. Is why plus one to the fifth Times F. Prime is 10 wide of the night. Now we subtract F times G brian After its wide at the 10th. G prime is five, Y plus one to the fourth. And then on our denominator we have G squared, which we said was why? Plus one to the 10th? Um So we have a little bit that we can do here. Let's first factor out our greatest common factor on top. Um So I see 5 to 10 so I can definitely take a five out, I have a wide of the 9th and a wide of the 10th, so I can take quite to the ninth out. And both terms also have that Y plus one factory in common and they both have four so we can take that out as well. So what remains? So from the first term I need a two and I need a Y plus one minus in the second term five is already taken out, I still need a why and that is all. So let's copy down our denominator five Plus 1 to the 10th. Uh so good news is I can make some cancellations, the swipe post under the 4th can cancel out And I can change this to a six. So I'm just subtracting for. Um So let's go ahead and simplify a little further. So I have five Y to the 9th. Let's go and expand these parentheses here. So this is two Y plus two minus Y. So in this parentheses I really have why positive and then on bottom we have Why Plus 1 to the 6th and this is about as simplified as we're going to get. So this is going to be our final answer.