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Evaluate the derivatives of the given functions for the given values of $x$. Use the product rule. Check your results using the derivative evaluation feature of a calculator.$$S=\frac{2 n^{3}-3 n+8}{2 n-3 n^{4}}, n=-1$$

Calculus 1 / AB

Chapter 23

The Derivative

Section 6

Derivatives of Products and Quotients of Functions

Derivatives

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All right, you're gonna want to have your calculator ready to evaluate D. S. D. N. When N equals negative one. Because if we noticed that s is a giant quotient. Uh and you're going to have to cube some things you have to divide. It's not the prettiest problem. Let's put it that way. three m plus eight All over to n minus three and To the 4th power. So as you notice, I mentioned the quotient in here, we're going to do the quotient rule. So uh stick with me. We're going to take the derivative of the top which is your power rule multiplied by the Coefficient with the exponent and then subtract one from the exponent. Leave the denominator alone minus Now take the derivative of the bottom. Again, I'm just following the power rules that we've already learned. Leave the top alone, all over the denominator two and -3 and to the fourth. And that quantity squared. Um so from here I know a lot of people would try to simplify, I wouldn't, I would just be plugging in negative one and for all of these ends I know there's a lot of them but stick with me. The reason why I mentioned that it's not that bad is because a few of the answers are going to be the same. Like negative one squared is just possible one, Six times one is 6 -3 is three. Um So again you have a negative one. So we're looking at negative two and negative one to the fourth, power is positive one. So you get negative five right there. But if you notice you actually get negative five the same term this is the same term as in the denominator. So that must also be negative five when you square it. now be careful because -1 when you cube it is a Still a negative. So that will turn that into plus 12. So we're looking at 14 because it's two plus 12. So here's another um -2 and becomes plus three plus eight. Uh So what are we looking at there? Nine? So as I simplify, I'm seeing negative 15, I don't trust my math to do 14 times nine in my head. It's -126, All over 25. And this is why I would use a calculator. Uh negative 141, divided by 25, Gives me an answer of negative 5.64. And as I check that it is correct, let's circle that and go onto the next problem.

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