Problem

Suppose $f$ is differentiable for all $x$ and con…

Problem 87 Hard Difficulty

Calculate the following derivatives using the Product Rule.
a. $\frac{d}{d x}\left(\sin ^{2} x\right)$
b. $\frac{d}{d x}\left(\sin ^{3} x\right) \quad$
c. $\frac{d}{d x}\left(\sin ^{4} x\right)$
d. Based on your answers to parts (a)-(c), make a conjecture about $\frac{d}{d x}\left(\sin ^{n} x\right),$ where $n$ is a positive integer. Then prove the result by induction.

Answer

a. 2 sin x cos x $\\$
b. 3 $sin^{2}$ x cos x $\\$
c. 4 $sin^{3}$ x cos x $\\$
d. $n$ $sin^{n-1}$ x cos x

Related Courses

Calculus 1 / AB

Calculus Early Transcendentals

Chapter 3

Derivatives

Section 5

Derivatives of Trigonometric Functions

Discussion

You must be signed in to discuss.

Top Calculus 1 / AB Educators

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 90

Video Transcript

So this question asked us to find the derivatives of various signed functions and part egg. We have sine squared of X and we can rewrite that a sign of X time, sign of X and we can define our A term as a sign of X and therefore a prime with equal co sign of X and E. We can define also a sign of X, which means be prime. I would also equal co sign of X. So now we saw this derivative. We can dio sign of X Times co sign of eggs plus sign of X Times co sign of X and because these are the same terms being multiplied together, remember that the multiplication rule says that we can switch these terms their order because they're always gonna equal the same thing. And because these are obscene terms, we get to close line of X sign of X. Part B now asked us to find the derivative sign, keep decks which we can rewrite as signed square to barracks times sign of X, And for this sine squared, we can use the chain rule to find its derivative and then plug that back into the product role so the chain rules formula is F prime g of x times g prime of X. We can define f of X to be you squared you being a placeholder for G and therefore a crime would equal to you and then g of X would equal sign of ex n g prime would equal Cosan. So the chain rule here would give us to sign of x co sign of X And now we can plug that back into you the product rule. So for the product role we know our A is equal to you sine squared x a prime is what we just found to sign of x co sign of X and B is just like before he would equal sign of x with be prime equaling co sign of X So now we can solve this derivative and we have to sign of x co sign of axe times Sign of X plus sine squared of x Times Co sign of X and notice how we have a sign of X in the sign of X here. So when we multiply this through, we'll get to sine squared of x co sign of X plus sign squared of X co sign of X And because these air like terms we can combine this Teoh equal three signs squared of x co sine squared of X and hurt See No, we have to find the derivative of sign to the fourth Power X and I hope you see the pattern here. But now we can rewrite this as sign keep of Eriks Times Sign of X And similarly, we're gonna want to use the chain rule for this term here so we can set f of X equal to you. See the three f prime of X with therefore equal three squared and G of X would equal sign que sign of x my bad sign of X and then G crime of X would equal co sign of X and the chain rule here. Well, give us three sine squared of x co sign of X. So again we can define her. A to be signed huge of X and a prime therefore equals three signed squared of x co sign of acts in This comes from this chain rule here B we considered equal to sign of acts would be prime Equalling co sign today and B equals co sign of X. So now we'll make do the product rule we have Sign Cube of X, My Bed. Wrong term three. Sign squared of x Co Sign of X Times Sign of X plus sign Cute of X times Coastline of X And when we multiply this through we get three. Sign Keeve of X Co Sign of X plus Sign Keyed of Acts co sign of X, and we can combine these terms to four Sign Cubed X Co sign Max, and if you haven't yet realized a pattern that's going on, it's OK, but notice where this four and this three is coming from. And if we look back at the original, we have this four, which is brought down in front of the sign, and we get three from four minus one. So therefore, for part D, these three dots mean therefore the derivative of sign to the end of X is equal to end times Sign of n minus one x co sign of X, and we can prove this with in example. So let's say we set an equal toe one. They would be finding the derivative of a D D X sign of X, which we get coastline of X and come sign of x a fancy way of writing. It would also be one coastline, X sign of one minus one x one minus one gives us zero and anything to the zero exponents always equal one. And we normally don't write the one constant. Um, what we normally don't write the constant if it's one, so you can see that it follows this form here.

Get More Help with this Textbook