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Numerade Educator

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Problem 21 Medium Difficulty

Find the derivative of the function.
$ y = \sqrt \frac {x}{x + 1} $

Answer

$y^{\prime}=\frac{1}{2 \sqrt{x}(x+1)^{3 / 2}}$

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Video Transcript

all right. Here we have a composite function and the inside function is a quotient. So we'll be using the chain rule to differentiate. But when we get to the derivative of the inside, we're going to use the quotient rule. And what I would like to do is when I have a square root, I'd like to write it as a 1/2 power. So we have X over X plus one to the 1/2 power. Now let's differentiate. So we're starting with the derivative of the outside. So we're going to bring down the 1/2 and raise a quotient to the negative 1/2 power. Now we're going to find the derivative of the inside. So this is where the quotient rule comes in. So we have the bottom X plus one times the derivative of the top one, minus the top x times the derivative of the bottom one over the bottom squared, X plus one quantity squared. Okay, Now let's see what we can do to simplify this. All right? There are several steps we can take here. First of all, let's work on this numerator. Ignore the times. One that doesn't do anything to us. So what we have is X plus one minus X. So that whole thing is just one. Now, thinking about the negative exponents and negative exponents means a reciprocal. So we can just put that fraction upside down and make it to the positive 1/2 power. So that means we're going to have as our numerator X plus one to the 1/2 power that comes from this part. And then we have a one from this part, and then we have a one from this part. So that's it. And then for the denominator, we have the two that comes from the 1/2. We have X to the 1/2 power that comes from this part. Remember, we took the reciprocal. So now that's on the bottom. And we also have X plus one to the second power, which was already on the bottom. Okay, so the last thing we could do then, is to remember that when we have the same base to a power divided by ah, that same base to a power, we can subtract the powers. Or we could think of it like this. You have an extra three halves of the X plus one quantity on the bottom. So let's cancel the X plus one to the 1/2 on the top of the bottom, leaving us with X plus wanted three halves on the bottom. And then we can also write the X to the 1/2 as a square root. So we have to square root X Times X plus one of the three hats on the bottom, and we just have a one on the top and there's are derivative.