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Find the equation of the tangent line to the curve at the given $x$ -value.$$y=\frac{\left(x^{2}+1\right)^{3}}{(x+1)^{4}}, \quad x=1.$$

$$y=1 / 2 x$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Missouri State University

University of Michigan - Ann Arbor

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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any time you're asked to find the equation of a tangent line, what you need is a point and a slope. Well, they give you half of the point because they give you the X coordinate is X equals one onda. We also need to slow, but I'll talk about both of those. Well, what we need to do is go to the equation, then X squared, plus one cubed over X plus one to the fourth. The first thing you would be to find the Y coordinate, and that's just plugging. Exit goes one. And for both of these exits, eso one squared is one plus one is two, so we have to cube over. One plus one is two to the fourth. I think it's pretty evident then that there's four twos on bottom and three twos on top of three of them. Can't cancel. So I left with one half of the point. Well, how do we find the slope? A slope would be the derivative, and what you need to do then, excuse me, is the change rule and the question Well, it's mainly a question rule, Um, because the big thing is being divided, but the change on top would be three in front X squared plus one is now squared. You leave the denominator alone. Whoops. I forgot to take the drift of the inside. I said it was a channel. But then I forgot to And then minus you leave the top alone X squared plus one, uh, cubes. And then the chain roll the bottom we bring the forefront X plus one cube all over the denominator squared. Well, if you take to the fourth power squared power to empowers multiply. So it's to the eighth power. And again you plug in one and for all of these excess. So that's gonna be a lot. Uh, what's nice is all of these because we did the math earlier. Whether it's X squared, plus one or X plus one, it's gonna be a bunch of twos. So we're looking at three times two squared times, two to the fourth times, two times two before minus two cube times, four times too cute. All over to to the eighth. Um, see, if I were doing this problem, I probably factor out the two to the six is they each have it. So we're looking at two to the six over, uh, C three times four would be left over 12 minus that four over to to the six times two squared. So that cancels that canceled. So I'm looking at eight. Divided by four, which gives me an answer of to which does not match the answer key. So I hope I didn't do anything. They took us one squared times the derivative of the inside. Leave the bomb alone minus, uh, leave the top alone for extra ones. Cube. Transmitter of the inside. I don't think I did anything wrong. The answer case wrong. Anyway, So I got the slope to be to, um maybe I made a mistake because the answer key says the answer is, Why equals one half X? But the answer I got would be white was too. That was the X minus one minus one has, which is the soap really was one half. It would simplify to this. Otherwise I got This is my answer. I don't see my mistake anywhere. Yeah, oops, Too far.

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