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 derivative at the given point on the curve using equation (2).$f(x)$ as defined in Exercise 4.

2

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Oregon State University

Harvey Mudd College

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.

02:57

Determine the derivative a…

00:51

Find the derivative of the…

01:05

Evaluate the derivative of…

01:01

01:02

02:20

Find a function $f$ with t…

01:51

Find the derivative of eac…

find the second derivative…

00:36

02:19

Find the second derivative…

01:32

all right. Using the function you see boxed in here, we're gonna be determining the derivative of F of X as given, which is equal to X squared plus three so that we can determine the slope of this curve at X equals one. To do this, let's start by breaking apart this equation in the box so that we can get each individual piece. Let's start with F of X plus age F of X plus. H is simply plugging X plus h in for anywhere. We have an X and R F of X. So this ultimately gives us X Plus h squared plus three. You can see that this expose ages just where X was. Now, if we want to simplify this, we can split apart our X plus h There we go plus three. And then let's go ahead and foiled this out. Distribute it so that we can keep it simplified. We end up getting X squared plus two x h plus h squared plus three. That's gonna be our first part of our functions. Now F of X is obviously just the ffx as given. So let's take what we found to be F of X plus h X squared plus two X h plus h squared plus three We're now subtracting or a FedEx which is X squared plus three Distribute this negative here, giving us X squared plus two X h plus H squared plus three minus X squared minus three. Now we see that we're able to cancel out our X squares and our threes, leaving us with two x h plus h squared. Go ahead and pull out that age, giving us two X plus age Now keeping in mind, we still have to divide by age as given in our function up here. I don't want to forget about that. We were just working with the numerous numerator. But let's add the denominator in now because we factored out that h on the numerator. We can cancel these off. This leaves us with two X plus h as our simplified function where we have f prime of X is equal to the limit as H approaches zero of two eggs plus h here we see we plug zero in for H. We're left with two x so that is our derivative here. But now we want to find the derivative at X equals one. So let's go ahead and do f Prime of one is equal to two times one giving us our answer where f prime of one is equal to two.

View More Answers From This Book

Find Another Textbook

Numerade Educator

01:35

Show the graph of the function defined by $f(x)=\frac{2 x-4}{x+3}$ does not …

02:22

a) sketch the graph of the given function, and then draw the tangent line at…

01:30

Use the vertical line test to determine if the given graph may represent a f…

02:08

Decide whether or not the function is continuous. If it is not continuous, i…

01:43

Find the equations of the lines and plot the lines from Exercise $47$.

02:55

Find the point on the curve $y=x^{3}$ at which the tangent line at (2,8) cro…

02:01

05:46

(a) Determine the equation of the tangent line to the circle $x^{2}+y^{2}=25…

02:50

In general, how are the lines $y=m x+b$ and $y-k=m(x-h)+b \mathrm{re}-$ late…

01:52

$$f(x)=\left\{\begin{aligned}4 x-2 & \text { if } x \leq 1 \\x+1 & \…