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Find the derivative of the function.$ g(u) = ( \frac {u^3 - 1}{u^3 +1})^8 $
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01:01
Frank Lin
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 4
The Chain Rule
Derivatives
Differentiation
Mario R.
February 24, 2019
Campbell University
University of Michigan - Ann Arbor
University of Nottingham
Boston College
Lectures
04:40
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
44:57
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
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here we have a composite function, one function inside another and the inside function is a quotient. So we're going to be using the chain rule and the quotient rule here. So to find the derivative, let's start with the chain rule. So the outside function is the eighth power function. So we bring down the eight and we raise the inside to the seventh. Now we multiply by the derivative of the inside. So here's where the quotient rule comes in. All right, so we have the bottom you cubed plus three times a driven her, plus one times the derivative of the top three. You squared minus the top u cubed minus one times the derivative of the bottom three. U squared over the bottom squared so over you cubed plus one squared. Okay, now it's all about simplifying this. So let's start with simplifying the numerator that we got from the quotient rule. I suspect that maybe some things you're going to cancel, so we'll just rewrite what we have. We have eight times you cubed minus one to the seventh over. You cubed plus one to the seventh. We can go ahead and split that up in anticipation of what might happen. And now we're going to simplify the numerator from the second fraction. So if I distribute the three you squared, I get three you to the fifth power plus three, you squared. And then if we distribute the minus sign as well as the three you squared, we get minus three you to the fifth power plus three, you squared and that is all over you cubed plus one squared Notice that you can cancel the three you to the fifth and the minus three you to the fifth and you can add three You squared and three year squared together and we get six years squared. So we're going to multiply that six You squared by the eight that we have out here and that's going to give us. I'm going to go over to the side here that's going to give us 48 u squared and in the numerator, we also have our you cubed minus one to the seventh and that whole thing is over. Notice that we have you cubed plus one to the seventh Power and you cubed plus one of this second power. So add those powers together and we have you cubed plus one to the ninth power. That's our derivative
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