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Numerade Educator

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Problem 26 Hard Difficulty

Find the derivative of the function.
$ s(t) = \sqrt \frac {1 + \sin t}{1 + \cos t} $

Answer

$\frac{\cos t+\sin t+1}{2(1+\cos t) \sqrt{(1+\sin t)(1+\cos t)}}$

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Video Transcript

here we have a composite function, one function inside another and the inside function is a quotient. So we're going to end up using the chain rule and the quotient rule. As we work through finding this derivative, I'm going to rewrite S of T as one plus sine of T over one plus co sign of tea to the 1/2 power. That way I can use the power rule in differentiating. So let's go ahead and find the derivative. So we're going to find the derivative of the outside function first, the 1/2 power function. So we bring down the 1/2 and we raise our quotient to the negative 1/2 power. Now we're going to find the derivative of the inside and we need to use the quotient rule. So we have the bottom one plus co sign t times The derivative of the top. The derivative of Scientist E is ko 70 course, a derivative of one is zero minus the top one plus sign T times the derivative of the bottom and the derivative of one plus coast 90 would be negative sign t over the bottom squared, so that's over one plus coastline t squared. Okay, now let's simplify. So when we have something to a 1/2 power, it means square root. But if we have it to a negative 1/2 power, the negative part means reciprocal. So what we could do is just find the reciprocal of this and write the square roots in there as well. So we have one over to Times Square root of one plus Coast, 90 on the top and one plus scientists on the bottom. And of course, we don't have to write the one that's not necessary. Okay, And then let's go ahead and simplify the numerator that we got from the quotient rule. We're going to distribute the co sign, and we're going to distribute the negative sign as well as the negative in front of that. And that's going to give us co sign t plus co sine squared T plus sign T plus science where t got to be real. Careful with those negatives there. Make sure you don't overlook any of them, and that's over. One plus co sign T quantity squared. Okay, now, remember your triggered a metric identities. Coastline square, T plus sine squared tea is What do you remember? Co sine squared data plus signs where data or tea or X or whatever you want to use in there is one so we can replace that with the one. So this point we have the square root of one plus co sign t over two times the square root of one plus sign T times one class CO sign T plus Cy Inti over one plus co sign T quantity squared. Now, this answer is looking great. But then you notice that you have ah one plus co sign t to the 1/2 power on the top and you have a one plus co sign t to the second power on the bottom and you realize that you want to combine those. So let's go back and let's right the one on the top that had a square root sign as one plus Coast 92 the 1/2 power. Maybe we prematurely changed it to a square root sign. So one plus co sign t to the 1/2 power. Okay, so now what we want to dio is a track. The powers. So we end up with the 1/2 power minus the second power is the negative three halfs power. So we end up with one plus ko 72 the three halfs power on the bottom. So let's go ahead and get some space to write that out. So this is what we saw in the last slide. So we're combining that term in this term by subtracting the powers and we end up with one plus co sign T plus Cy Inti over two times the square root of one plus sign T Times one plus CO. Sign T to the three house power. And then we say to ourselves, Well, this is inconsistent. We have one of them with a square root sign and one of them not with a square root sign. So I say, That's okay, we can change that. So we have one plus co sign T plus I Inti, over two times the square root of one plus sign T times the square root of one plus co sign T cubed. And then we say to ourselves, Well, we shouldn't keep something cubed inside a square root sign. So let's simplify that. So now we have one plus co sign T plus sign T over two times and we factor out a one plus co sign t. And so we have the square root of one plus sign tee times one put, uh, Times one plus co sign t. We could put them in the same square root sign, couldn't we? So if we do that, we'll call it done. So let's put these inside the same square root sign. We have the square root of one plus sign T Times one plus CO sign T. Now remember that you may not be required to do all these steps of simplifying. It kind of depends on your instructors preferences. And there are not hard and fast rules about simplifying. So what I'm showing you is how to get it down to the point where it looks like the answer that they had in the book.