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Find $d w / d r$ if $w=\left(5 r^{2}-2\right)^{2 / 3}\left(r^{2}-1\right)^{1 / 2}$.

$$\frac{r\left(35 r^{2}-26\right)}{3 \sqrt{r^{2}-1}\left(5 r^{2}-2\right)^{1 / 3}}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

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University of Nottingham

Idaho State University

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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we have to do the product rule and the channel to do this problem. Five R squared minus two to the two thirds power. And I'm gonna multiply that by r squared minus one to the one half power. So the main thing is, we have this product in between. So that tells us we need to do the product rule and the chain role because, as I do the WDR, the first thing in the product cools Take the derivative of the left side, which is the change. Will you gotta bring that two thirds in front? Five R squared minus two stays the same and you subtract one to the exponents. So that's negative. One third, um then the multiplied by the derivative of the inside, which is 10 are and then multiply by the right side of the product that we haven't touched yet. You leave that alone then plus you leave the left side alone to the two thirds power and then you take the derivative of the right side, which is the product rule. Bring that one half in front R squared minus one is now to the negative on half power times. The derivative of the inside, which I was just right right here, because So what happens? You multiply, goes into the numerator or something. Now, some teachers and I'll include myself in here. We'll let you leave your answer like this because, you know, what's the point of simplifying? Uh, you know, when do you stop? I guess other teachers will make you work through the problem on it breaks down to our 35 r squared minus 26. Yeah, all over. Because we have all these negative exponents, I would move into the denominator. Three R squared minus one is now to the one half power. And then that five r squared minus two is to the one third power. And, uh, some teachers will make you work through the problem and simplify until you get here. So, uh, simplifications up to your teacher or your preference

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