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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.

Calculus 1 / AB

Chapter 23

The Derivative

Section 7

The Derivative of a Power of a Function

Derivatives

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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'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's precisely what I'm gonna do here. So I'm gonna rewrite Why as gonna be to root X minus one times and then this is over one. I'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'm gonna rewrite their squared of Axa's extra the 1/2 power minus one, divided by four X plus one. So now I didn't do any calculus yet. This is just the same function as I started with. But now it's written in a form where we can used a portion role and we shouldn'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's driven of zero, so it doesn'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're X plus one times one acts to the negative 1/2 power. Okay, now the derivative this Since this is linear, it's just the slope of this line, which is for I'm actually taking four times two x to the 1/2 minus one and dividing by. We're explosive. Once word now, I'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'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's gonna cost problems for your four. So here you go.

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