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Find the equation of the tangent line to the curve at the given $x$ -value.$$y=\left(x^{2}-1\right)^{3} ; \quad x=2$$

$$y=108 x-189$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Campbell University

Oregon State 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.

04:05

Find an equation of the ta…

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03:34

Determine the equation of …

04:22

The overall theme of a tangent line problem is to make sure that you have a point and a slope. Well, finally, the point in a problem is not too bad, as long as you know the X value, which they tell you the exciting this problems too. So let's look at the problem. Now we have y equals X squared minus one cube. Well, if you plug to infer that X, you'll figure out what why is to square this four minus one is, uh, three three cubed is 27 or you can use a calculator, not there like that. But then the second part. Well, how do you find the slope? Well, you need the derivative Do the school y de X at that ordered pair or or in this case, when X equals two. Because now we're just going to use the chamber. It's a general. Well, you bring the three in front, you leave X squared minus one alone. It's now to the second power, and then you multiply by the derivative of the inside, which is two X and again we need to plug to and for both of these excess, what's nice? Those we already did. Two squared is four minus. One is three. Three. Squared is nine times this. Three. It's still 27. It's Write that down. Um, I missed rank two times. Two is 4. 27 times four. Uh, let's see, 25 times four is 100 and then two times 48. So it must be 100 and eight. Now, how would I write? My tangent line is I am a big fan of point. So form where I would write why 108 x minus the exponents plus the y coordinate. Uh, you don't. I mean, that this is one option. This is a correct answer. Or you could distribute that 108 end 108 x of U minus, 216. But then you have to combine like terms. So that's combining these two things together, which would give you why equals 108 x. I don't know why people like slope intercept form, but they do. We'll give you this answer, but it's up to you. Do you like this answer more or this answer more

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