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

Repeat Exercise 16, for: (a) $f(x)=x^{1 / 2} /(x+1),$ (b) $f(x)=(x+1) / x^{1 / 2}$

(a) $(1,1 / 2)$(b) (1,2)

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

00:24

$$f(x)=(x-2)^{2}+1, \q…

01:09

$$\text { Find } f^{\prime…

19:22

$F\left(\frac{1}{2}, 1 ; \…

Let $f(x)=2^{x} .$ This fu…

02:08

All right, So we're looking at these reciprocal functions X to the one half over X plus one are gonna do the math there and then on this side we're gonna do X plus woman over X to the one half. So they're definitely reciprocal functions. We're talking about horizontal tangents. Eso When you see that phrase horizontal tangents, you should automatically think Okay, if this is f of X equals than what we need to do is find the derivative and set that equal to zero. And that Onley happens when the numerator equal zero. We don't care about the denominator. Um, yeah, so we're gonna do the quotient rule, which is a derivative of the top is one half X to the name of one half power. You leave the bottom alone minus. Now you take the directive of the bottom leaves the top alone all over the denominator squared. But again, we don't really care about the denominator, because that, I mean, that could never even equal zero because we're only going to get positive numbers there. I guess it could equal zero with excess negative one. But anyway, we don't care about the denominator. So as we examine this problem and we distribute in the numerator, Uh, we start adding things. We have one half X to the one half power plus one half X to the negative one half hour minus X to the one half Power is equal to zero. Um, would see one half minus one. Uh, would be negative one half. And I want to add that over. So the equals, uh, one half X to the negative one half hour. Well, these twos could cancel. Um, so we're looking at one over Route X equals route X. So then multiply that over. I'm gonna get one equals X. There you go. Which matches our answer? That X equals one. There you go. On the underlying that because that's not technically your final answer. You have to go back to the original plug in one eso the ordered pairs wine to the one half hour. Still one and one plus one is two. So that's the first point. And as we look at this one, really, the work is the same because the derivative I wish they renamed it something other than F. But it's really the same work driven to the top to the bottom alone, minus the derivative of the bottom. Leave the top alone all over the denominator square. And so it becomes extra the first. So if you set the numerator equal to zero when you start distributing in here, what you'll see is that you have X to the one half minus one half x to the one half minus one half X to the negative one half and again equals zero. And as I combine like terms So I did over here and then get rid of the twos. You'll see you end up with the same equation. One overruled X equals route X. So that means that X has to equal one. As you go back to the original problem and you start plugging one in for X. Well, one plus one is two and one to the one half hours one and two. Divide by one is one. Every sorry is too. There's your other answer. We did it

View More Answers From This Book

Find Another Textbook

05:35

Determine the equation of the tangent line to the given curve at the indicat…

01:16

Suppose a function is continuous on $[a, b]$ and differentiable on $(a, b) .…

04:01

The number, $N$ (in million) of VCR's sold in the United States for the…

01:10

Determine whether or not the function determined by the given equation is on…

01:34

Find the instantaneous rate of change of $y$ with respect to $x$ at the give…

05:05

The area of a square is increasing at the rate of 1 square inch per minute. …

02:03

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

01:33

If $f(x)=a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x^{1}+a_{0} .$ Find the f…

08:06

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

01:20

Sketch the graph of the function defined in the given exercise. Use all the …