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Numerade Educator

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Problem 34 Hard Difficulty

Find the derivative of the function.
$ U(y) = (\frac {y^4 + 1}{y^2 + 1})^5 $

Answer

$\frac{10 y\left(y^{4}+1\right)^{4}\left(y^{4}+2 y^{2}-1\right)}{\left[y^{2}+1\right]^{6}}$

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Video Transcript

here we have a composite function, a function inside of function, and the inside function is a quotient. So we're going to end up using the chain rule to take care of the composite and the quotient rule to take care of the derivative of the insect. So here we go. So first of all, the derivative of the outside function, which is the fifth power function, would be to bring down the five and then raise the inside to the fourth. Now we're going to multiply that by the derivative of the inside. So here's where the quotient rule comes in. So we have the bottom y squared, plus one times the derivative of the top for why cubed minus the top. Why did the fourth plus one times the derivative of the bottom two? Why over the bottom squared, Why squared plus one quantity squared. And now we're going to simplify. Okay, so I'm gonna write this quotient here as the top to the fourth over the bottom to the fourth with five times. Why did the fourth plus one to the fourth over? Why squared plus one to the fourth? And I do that because I suspect that some things were going to simplify as we go. Some things you're going to cancel. For example, we can already see that the denominator of the second fraction has some terms that will cancel with the numerator of the first fraction we're actually, they will melt. They will combine with the denominator of the first fraction. Excuse me. Okay, so now we want to simplify our numerator of the second fraction. So we're going to distribute the four y cubed, multiply it by both of those terms and distribute the to y multiplied by both of those terms and distribute the negative. So we have four wide of the fifth power plus four y cubed minus two y to the fifth power minus two. Y all right. Now let's see what we can do with that. Notice that we have four white of the fifth minus two white of the fifth so we can combine the like terms and we get to y to the fifth, and we're gonna have to create a little bit more space to work here. All right, so this is what we just saw a moment ago. And what we're doing now is we're combining the four white to the fifth, minus two y to the fifth. So now we have five times y to the fourth plus one to the fourth times two white of the fifth plus four y cubed minus two y over. Now let's go ahead while we're rewriting this and combined these into y squared, plus one to the six power Okay, finally, let's notice that each of the factors for each of the terms in the the final factor of the numerator can eyes a multiple of two y. So let's factor to why, out of each of those and we'll multiply that to why that we get factored out by the five. So that will give us 10 y times a white of the fourth plus one to the fourth times. Why did the fourth plus two y squared minus one and that's still over? Why squared plus one to the six power