Problem

Find the derivative of the function. $ r(t) = 10…

05:04

Problem 26 Hard Difficulty

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

Answer

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

Watch More Solved Questions in Chapter 3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70

Problem 71

Problem 72

Problem 73

Problem 74

Problem 75

Problem 76

Problem 77

Problem 78

Problem 79

Problem 80

Problem 81

Problem 82

Problem 83

Problem 84

Problem 85

Problem 86

Problem 87

Problem 88

Problem 89

Problem 90

Problem 91

Problem 92

Problem 93

Problem 94

Problem 95

Problem 96

Problem 97

Problem 98

Problem 99

Problem 100

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.

Frank L.

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Problem

Find the derivative of the function. $ r(t) = 10…

Problem 26 Hard Difficulty

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

Answer

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

More Answers

Related Courses

Calculus: Early Transcendentals

Related Topics

Discussion

Top Calculus 1 / AB Educators

Grace H.

Catherine R.

Kristen K.

Joseph L.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 3

Video Transcript

Calculus: Early Transcendentals

Catherine R.

Kayleah T.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview