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Question

Solve the given problems by finding the appropriate derivatives.
Find any values of $x$ for which the derivative of $y=\frac{x}{\sqrt{4 x-1}}$ is zero. View the curve of the function on a calculator to verify the values found.

Alex Roush · Accepted Answer

Okay, so we have to find the derivative of this function. Now, at first glance, it appears that we have the product to functions, so the product rule would make sense. Except we actually don&#x27;t know how to differentiate something like this. You have to learn the chain rule in order to do that. But for right now, we have to turn this into something else that we do know how to take the derivative. So remember that if I have something like X rays to the negative first power, this is really X divided by one. And properties of exponents states that you have to flip this and then you can turn this one this negative one into a positive one. So you can foot the fraction as long as you change the sign of the exponents. Yes, that&#x27;s precisely what I&#x27;m gonna do here. So I&#x27;m gonna rewrite Why as gonna be to root X minus one times and then this is over one. I&#x27;m gonna flip this and raise everything to the first power, which is the same as doing nothing to it. Right. Okay, so this is actually written as two. And then I&#x27;m gonna rewrite their squared of Axa&#x27;s extra the 1/2 power minus one, divided by four X plus one. So now I didn&#x27;t do any calculus yet. This is just the same function as I started with. But now it&#x27;s written in a form where we can used a portion role and we shouldn&#x27;t get into trouble. Okay, so, uh, why prime is therefore going to be bottom function times the derivative of the top minus the top function times the derivative of the bottom function, all divided by the bottom functions squared. Okay, now we just have to take a couple of derivatives. The first derivative is just one generation of the power, all right, because with a derivative of a difference or some, you split this up into the derivative of this minus the derivative of this. But what is a constant and it&#x27;s driven of zero, so it doesn&#x27;t really account for anything. So here we use the power rule, we bring this exponents down is a coefficient. So you take two times 1/2 to get one, and then we decrease this by one. So 1/2 minus one Same as 1/2 minus 2/2, which is minus 1/2. Okay, so Miss Derivative tells us that white prime is equal to we&#x27;re X plus one times one acts to the negative 1/2 power. Okay, now the derivative this Since this is linear, it&#x27;s just the slope of this line, which is for I&#x27;m actually taking four times two x to the 1/2 minus one and dividing by. We&#x27;re explosive. Once word now, I&#x27;m gonna go ahead and distribute this term, see what we get. So I get a four. Now, this is raised to the first power, and this is to the negative 1/2 power. And when you multiply, when you multiply these guys and they have different exponents, as long as they have the same base, then you add them. You&#x27;re taking one plus a negative 1/2 which just gets you Ah, positive one house. And then the second term distributes to get you next to the negative 1/2 minus eight extra. The one hour plus four. Okay, so my last step is gonna be combining those x to the 1/2 powers. So I have four minus eight of them, which combined to get Mia negative for next to the 1/2 plus extra the negative 1/2 plus four, all divided by four x plus one squared. And to me, this is simplified enough Can factor and X to the negative 1/2 out if you want, But that&#x27;s gonna cost problems for your four. So here you go.

Amy Jiang · Answer

Okay, so we see that we have a product of two transparent. So when we take the derivative, let&#x27;s use our product rule. Just, um, two hands. One a half. So that&#x27;s this extra part of negative 1/2 times for X +12 part of negative one and then plus distributive off this term here, that&#x27;s negative or X plus one. That&#x27;s part of negative one of minus wanted it to go to an end Times conservative of what we have are Okay, so we get this as our derivative.

Karl Schaefer · Answer

and this problem, we&#x27;re given the function. Why equals one minus one over X ratifying D Y d X at X equals three. So all this all this notation means is this is just the derivative at X equals squared of three. So Step One is to find the derivative find D Y D X, and Step two is to plug in X equals squared of three. So do I. D X, this is just the limit. The limit is h goes to zero. Now we just need to plug in X plus h So one minus one over X plus age minus one minus one of rex all over age. So we immediately we just see that we have some some cancellation here. So here&#x27;s a one hears minus one and then here&#x27;s minus a minus one over X. So we&#x27;ve got one of Rex minus one over X plus h over age. So here we&#x27;ve just combined the numerator to get a one over X minus one over X plus agent. We cancelled the ones. Now let&#x27;s go ahead and turn this fraction of the numerator to a single fraction by getting a common denominator. So women his age goes zero. We&#x27;re still all dividing by H, but in the numerator, we now has X plus age over x times X plus age minus X over X times X plus h Come appear. So this is the limit is H goes to zero of X plus age minus X over axe times X plus age all over H and the next step. Let&#x27;s do a couple things. So first I want to combine the two denominators into a single denominator by multiplying them together. And secondly, I want to cancel on the numerator X minus x so are left with this expression. But now we can cancel the ages. So we&#x27;re just left with the limit is H ghost zero of one over x times X plus h But now we can just plug in eight people zero. Now that we&#x27;ve got rid of the H and denominator, we can just plug in h equals zero and we get one over x squared. So this was our derivative, our overall derivative. But now we wanted to plug in X equals square root of three. So once we plug in X equal spirit of three, we get one over the squared of three squared, which is one third. Your final answer for the problem is one third

Ernest Castorena · Answer

function that&#x27;s right ends up term well after a over to here well of X to the 1/2 year and let&#x27;s see extra the negative 1/2 year. So if you take a pyro term by term, we&#x27;re gonna end up with Let&#x27;s see three house times for that&#x27;s six x to the 1/2 when we have minus 1/2 X to the negative one, huh? And lastly, we have my mystery over to okay, over to X to the negative three house.

Samantha Lincroft · Answer

So for this problem, were asked to find the derivative of the function on the left, using the quotient rule on the right. And so we&#x27;re going to be calculating Devi over the X and we&#x27;ll start by doing that as g times of crime or the bottom x times the top, uh, the derivative of the top, which will be one minus to overrule X. And then we&#x27;ll do that minus f, which is the top times the derivative of G, which is actually just one and then finally will divide that all by the bottom, which is X squared. And when we simplify that, we will get that the derivative is to root X minus one over X squared.

Problem

Solve the given problems by finding the appropria…

Problem 40 Easy Difficulty

Solve the given problems by finding the appropriate derivatives.
Find any values of $x$ for which the derivative of $y=\frac{x}{\sqrt{4 x-1}}$ is zero. View the curve of the function on a calculator to verify the values found.

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Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus