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

$3 / 17$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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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Determine the equation of …

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Find the equation of the t…

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here we have a function which we would like to find the tangent line to. It at X equals to. To do this, we'll be using some product rule quotient rule. Some rules a lot of different derivative rules in order to find this. But to begin, let's identify the fact that because we're working to find the equation of a tangent line, which is always going to be linear, we're eventually going to be working with point slope form, which you can recall Y minus y one is equal to m times X minus X one, meaning we need to find each of our variables here. So we were already given our X value. That's pretty straightforward at X is equal to two. To find Ry one, we just need to plug to into our function. So doing f of two, we find that Y is equal to zero. That gives us our coordinate point. Now we just need to find our slope, and our slope will be when F prime is equal to two. So we need to find our derivative and then plug in our X value of two. To start here, we're going to need to be using our product rule. You can see that in the numerator is two terms multiplied by each other. Product rule. I like to think of it as being F prime G plus g prime f. You can see near the function my first term. I've identified as F my second term. I've identified as G So doing this. Let's take the derivative of F, which is just one multiplying that by R G, which is X squared minus three X plus two. We're going to add the product of G prime. So taking the derivative G, we get two X minus three multiplying that by R F, which is just X plus one now this year would give us the derivative of our numerator. We can simplify this a little bit and find this we get X squared minus three X plus two plus two X squared plus two x minus three X minus three. Further simplifying. We have three X squared minus four X minus one. So this is in the whole scheme of things here. This is going to be our f prime Now that we're starting to work with our A quotient rule the quotient rule is very similar to the product rule where we have f prime G. But we're gonna subtract G Prime F. And in this case, our entire numerator is F and our denominator is G. So we just found f prime right here. So we have three x squared minus four X minus one that gets multiplied by G, which is our denominator X to the fourth plus one. Now we need to subtract R g prime. So taking the derivative of our denominator, we get four x cubed, multiplying that by our f which is our entire numerator giving us X plus one times X squared minus three X plus two. Now that we have this, we can start working towards simplifying it, which means really just multiplying all of these out. This is gonna give us three hopes giving us three x to the sixth minus four x to the fifth minus X to the fourth plus three X squared minus four X minus one minus All of this. Still, we still have four x cubed that's going to be multiplied by after we multiplied all of this out we get X cubed minus three x squared plus two X Plus X squared, minus three X plus two. We'll go ahead and now multiply everything we have in parentheses here by negative four x cubed, giving US three x to the sixth minus four x to the fifth minus x to the fourth plus three X squared minus four x minus one minus four x to the sixth plus 12 x to the fifth, minus eight x to the fourth minus four x to the fifth plus 12 x cubed minus eight X. Now that's a lot. So let's simplify even further, which will give us our f prime of X. So the derivative of our function is that negative. X to the sixth plus four x to the fifth minus nine x to the fourth plus four x cubed plus 15 X squared, minus four X minus one. All right, so now that we have our derivative, we can start, we can find our m in our point slope form, which will just be like we said before f prime of two doing that, plugging to in for all of our X values. Here you'll get negative 27 which is equal to M. Now let's start plugging this into our point slope form. So we have y minus ry one which we found with zero is equal to M, which is negative 27 times x minus R x one which is to solving for y. Here we end up getting y is equal to negative 27 x plus 54 that would be the equation of our tangent line to the function at X is equal to two.

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