Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Determine the equation of the tangent line to the given curve at the indicated $x$ -value.$$f(x)=x^{2}\left(\frac{x^{4}+3 x^{2}-8}{x^{3}+12}\right) ; \quad x=2$$

$y=52 / 3 x-84 / 5$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Campbell University

Harvey Mudd College

Baylor University

Boston College

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.

05:24

Determine the equation of …

04:06

02:26

03:00

02:14

Find an equation of the st…

02:10

02:47

Find the equation of the t…

02:20

02:05

02:39

Find $f^{\prime}(x)$ and f…

02:44

Find the equation of the l…

02:17

02:51

So first we want to find the derivative of this whole function And it's pretty lengthy one. So they're with me. But basically we have a quotient and a product. So first let's label are expressions we're gonna call the top one, you The bottom one v. This whole thing, the whole quotation we'll call Z. And then this product or this expression X squared will call W. And they're just arbitrary letters. I'm just labeling them. So you know what I'm taking the derivative of at any given time. So first we're going to work with our quotient expression which is the so D. Z over D. X. Which is the derivative of this entire quotient will be D. You over D. X. Times B minus DV over DT eggs times you all over he squared. So that what that means is we take the drift over the top function or the top expression multiplied by the bottom expression. To the director of the bottom expression. Multiply by the top expression and then all over the original bottom expression. So what that will look like is put it down here. The drift of the top function. Using power rule we have four x cubed which takes care of this term and then was six X. And then we have the multiply that by The entire 2nd expression which is x cubed Us 12 or the bottom expression fee. And then we subtract that with our entire topic expression which is export Plus three x squared minus eight. And multiply that by the derivative of the bottom function which using power rule 12 is a constant. So we don't take the derivative of that is three X squared. Running out of room already. And then all of that over X cubed plus 12 squared. So that is our quotient derivative. Now we work on the product rule with W. So our product rule is D. W. Or D. X times Z. In this case are quotient plus easy over D. X times W. So what that will look like? We take the derivative of the first expression X squared. And that would give us two X. Multiply it by the entire 2nd expression which is the quotient next fourth plus three, X squared minus eight Over x cubed plus 12. And we add that to the derivative of the second expression which is the question which we already did here. This is the entire derivative and then we multiply that by X squared. So I'm just going to clean this up a little bit. I'm not going to simplify and you'll see why in just, okay, so now that we have our expression, I didn't simplify it because what we're gonna do next is this X value right here X equals two. We're going to plug that in both into our affects and to our into our D. F. Effexor are derivative function. So what this will do by plugging your X. Value into ffx, You get the Y value at that at X equals two. And by plugging it into the derivative function you actually get the slope of the line that will cross at X equals two. So for after you like in x equals two to your app of X. So F. Of X equals two. And then your F. Of or start your F. Prime of X equals two. Or your derivative. You should get 52 3rd. So that is your slopes. Because we're trying to find the equation of the line right? And you're lying function is Y equals mx plus. Must be. So you get you got your wife, your M. Right now which is 52/3. And now we're trying to find your wife. So that's why you plug in your ex back into this original equation. And with that you should get. So once you get your Y. Which I got four, we have your Y. You have your M. And you plug an accent. You here to find your B. So Your final answer should be 52/3 X minus pay for this because why?

View More Answers From This Book

Find Another Textbook

01:46

Find two functions $f$ and $g,$ whose composition $f(x)$ ) will result in th…

02:45

Find the point(s) on the curve $y=6 x^{1 / 3}$ at which (a) the slope is $1 …

01:21

Use your knowledge of the derivative to compute the limit given.$$\lim _…

06:45

The Amalgamated Flashlight Company shows a profit of $4,500$ on a production…

01:06

Locate all critical points.$$f(x)=4-x^{2 / 3}$$

02:38

(a) Find the $x$ -intercept(s); (b) Find the vertical asymptotes; (c) Find t…

00:57

$$\text { Find } f^{\prime \prime}(x) \text { if: (a) } f(x)=x^{5}+3 x^{2}+5…

02:13

Find the equation of the line perpendicular to the tangent line to the curve…

04:08

The cost, in dollars, of producing $x$ bicycles is given by $C(x)=60+$ $10…

01:16