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If $f^{\prime}(u)=\frac{u^{2}-1}{u^{2}+1},$ and $y=f\left(x^{3}\right),$ find $d y / d x$.

$$3 x^{2}\left(\frac{x^{6}-1}{x^{6}+1}\right)$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Harvey Mudd College

Baylor University

Idaho State University

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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the way I would work through this problem is to figure out what why is first on as you look at it is defined as f of X cubed. So as I look at that problem and I do D y d x well, that tells me that what I need to do is the derivative of F leave X cubed alone and then multiplied by the derivative of the inside, which should be times three x squared. And then I would go back and use the definition that they tell you, which is that f prime of you. So this is given in the directions is equal to use squared minus one or you squared plus one. So as I go back to this problem, uh, f prime is defined as something squared minus one and something squared plus one. Well, what I need to do is replace that you and the problem here and here with execute, because that's from right here. And then I had still have that times three x square. Now, usually how we rewrite this problem. Just throw all of this in parentheses. Ah, lot of teachers will put that three x squared in front, but you don't have to do it that way. You could even put that into the numerator, and you have a power to empower. You multiply the excellence. It's X to the six minus one over X to the six plus one, and something like this is all we need. That's correct.

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