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Find the equation of the tangent line to the curve at the given $x$ -value.$$y=(x+8)^{2 / 3}\left(5 x^{3}-7 x+16\right)^{1 / 2} \quad x=0$$

$$y=\frac{-13}{6} x+16$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

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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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a soon as you see tangent line. What you should be thinking about is can you find the point and the slope on what's nice about this problem is they give you that the X coordinate zero and sort of the definition of exporting. Being zero is they're telling you what the Y intercept is when you find that part of the problem. Well, how do you find the Y intercept? And you have to look at the equation. Uh, oops. Forgot to write X X plus eight to the two thirds power Times five X cubed minus seven X plus 16 to the one half power. Uh, eso as I mentioned X zero. They tell you that in the problem. So we're gonna plug that in for all the zeros. And this is really nice because zero plus anything is that thing. So looking at eight to the two thirds power, which means the cube root of being squared. The same thing here is that zero minus zero plus 16 and then one half hour means the square root. And I would expect my students to know the square root of eight is to Cuba Motivators. Two squared is four on the square root of 16 is four. So the four times for 16 we already found the Y intercept. Well, how do we find the slow called The Slope represents the derivative when x zero s. So what we're gonna do is the product rule and the chain rule. In this problem, the product will because we have a product here, you bring that two thirds in front. X plus eight is now to the negative one third power. Uh, and you take time to during the the inside, which is just one. So no need to write times one there to one half hourly, the right side alone. And then you leave the left side alone to the two thirds power. Then you take the derivative of the right side, which is a chain rule. I've x cubed minus seven x plus 16 is to the negative one half hour because you subtract one from you. Explain it times the derivative of the inside, which would be 15 x squared, minus seven. And now you have to plug in zero for all these exes. There's a lot of zeros. And that's why they chose an easy number because then you're just looking at two thirds zero plus eight is eight. The negative exponents puts into the denominator. The Cuba debate is too. We already talked about it. 16 to the one half power is square root of 16, sometimes four. Andi already did eight to the two thirds powers four times one half the negative exponents on that one puts into the denominator. 16 square to 16 is four and then zero minus seven. Negative seven. So as we break this down, we switch to Green is thes to shoot. Cancel. We're looking at four thirds. Let's see this four can cancel this sort of again. Minus seven halves. If you get the same denominator over six. Let's see. This would be 86 Minus. Was that 21 6 would give me, uh, eight minus 13. Yeah, ive 13 6. Here we go. So that represents your slope? Um, e don't want to circle that, though. So we write our answer as the sloping of 13 6 x plus. The Y intercept we found earlier was 16. There we go. That is the equation of attention line

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