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.$$f(x)=\left(x^{4}-2 x^{2}+3\right)\left(x^{3}+1\right) ; \quad x=1$$

$y=6 x-2$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Missouri State University

Campbell University

Baylor University

University of Nottingham

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 …

02:10

04:02

Find the equation of the t…

02:37

04:22

Find an equation of the ta…

01:53

we have another tangent line problem, which means we need a point and a slope, and they give us the X coordinate is one. So we have to figure out the Y coordinate and the slope is found by taking the derivative and plugging one in for X. So as we look at this problem, X to the fourth one is two x squared plus three, uh, and X cubed plus one. Oh, what I would do is start by plugging in one and for all those values. What's nice about this, though, is one cubed is one plus one is two. And then, as I'm just looking at this value, I've seen one minus to donate. One plus three is 22 times two will give me a Y value. Four eso. Now I'm ready to find the derivative, which is your product rule. Take the directive of the left Side or X cubed minus four X. Leave the right side of own Q Plus one and then plus leave the left side alone and then times the directive of the right side. Now it's nice about this is where he figured out that X cubed plus one equals two. And then this piece also equals two. Is that to figure out that when we plug in one for these values, we get zero, You know, four times one minus four times one is going to give me zero. So zero times two plus three times one. Um Okay, so that's point. That's what we get is a slope of six to top 36 eso. Now we're ready to find the equation. I like point slope form. So I'm writing why it was a slope X minus the x coordinate. Plus or why coordinate? So I like this answer. But other teachers might ask you to distribute. If you distribute, you'll get six X, uh, B minus six, but plus four would give me minus two. So this is slope intercept form, and they're both correct.

View More Answers From This Book

Find Another Textbook

02:15

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

01:04

Sketch the graph of the function in Exercise 41

08:06

(a) John is at $B$, on a straight beach, 10 miles from $A$. Mary is in a boa…

04:33

A kite maintains a constant altitude of 50 feet and is moving horizontally a…

The price of a magazine, in cents, is given by the equation $p=10(147-0.02 x…

01:38

$f(x)=\left(x^{2}+3 x+1\right)^{17},$ find $f^{\prime}(x)$.

01:36

Suppose $f$ is not a constant function with $f(x+y)=f(x) f(y),$ show (a)…

01:57

A 12 inch piece of wire is to be cut into two pieces. One piece is to be use…

02:03

A holding pen for fish is to be made in the form of a rectangular solid with…

02:22

$$\text { Find } \frac{d^{2} y}{d x^{2}}, \text { if } x^{2}+y^{2}=1$$