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Determine the equation of the tangent line at the indicated $x$ -value.$$y=\sqrt[3]{x}+2 x-20 ; \quad x=8$$

$$y=\frac{25}{12} x-\frac{56}{3}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

Harvey Mudd College

Baylor University

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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So we're given another tangent line problem. And anytime you see Tangent Line, you should think I need a point and I need a slope. Well, the only thing that they give you in this problem is that X equals eight. Um, I'm going to write eight down for my excrement, but they don't give you the y coordinate it. And in order to find the slope, you need to find the derivative and plug eight into the problem. So as you look at the original problem actually makes more sense to rewrite the cube root as to the one third power plus two x minus 20. Um, now a t least I expect my students to know that this is the same thing as the cube root. Because when you plug in eight and I expect my students to know the Cuban debate is too plus 16, 2 times eight, that's 18 minus 20 will give me negative too. So that's my y. Coordinate. Well, the find your, um yes, I didn't realize that's why Equals? Well, the notation for this I'm gonna change. Why equals toe ffx? So then it matches my notation, but, um, it's the same thing. Finding the derivative is one third x to the native. Two thirds power plus two. Um, and finding this derivative is not not that bad. You bring the exponents front, subtract one from it in the derivative of a constant zero and now plugging eight into these values. And let's see, the cube root of eight is two. Squared is four Onda negative puts into the denominator. So three times four gives me 12 plus two. Let's see, that would be 24. 12 is equal to two. So we're looking at 25 12 as our slope. That's what the derivative tells us now. I would just leave my answer as why equals the slope and then x minus My x coordinate might, plus my wife, Cornet, which turns into minus two. I let my students leave my answer like this. I know other teachers will actually require you to distribute that in there and combine like terms. And if you're professor does that, then you finally answer would be 25 12 ex. I'm just using a calculator minus 56 3rd. There you go,

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