Question

Find the derivative of y = cosx / 1+sinx

I used the quotient rule using the derivatives of cos and sin but it just turned into one big equation. Can anyone help?

Answer 1

@cwrw238 Do you think you could help me with this?

Answer 2

Why would you use the quotient rule?

\(y = \cos(x) / 1 + \sin(x) = \cos(x) + \sin(x)\)

\(y' = -\sin(x) + \cos(x)\)

On the other hand, if you remember your Order of Operations, and you REALLY mean:

\(y = \cos(x) / (1 + \sin(x)) = \dfrac{\cos(x)}{1+\sin(x)}\), then that would be a Quotient Rule.

Show your work. Someone will help you with the calculus, now that the algebra is straightened out.

Answer 3

You may also wish to remember your trig identities.

Answer 4

How did you go straight to -sinx+cosx ?

Answer 5

I was looking at the trig identities but couldn't find anything to fit in the original equation.

Answer 6

You are not paying attention. I worked the problem you WROTE. I did not work the problem you INTENDED.

Answer 7

[ -sin x ( 1 + sinx) - cos x ( cos x)] / ( 1 + sin x)^2

Answer 8

@cwrw238 That's what I did and then expanded the equation and tried to simplify by using the half-angle formulas. Should I just leave it as it is?

Answer 9

\(\sin^{2}(x) + \cos^{2}(x) = 1\)

Half angle? Why?

Answer 10

= [- sin x - sin^2x - cos^2 x ]/ ( 1 + sin x)^2

Answer 11

yes you can use sin^2x + cos^2 x = 1 to simplify it further

Answer 12

This is what I got after using the quotient rule and half-angle formula.

\[y=\frac{ -\sin(x) + \cos(2x) }{ (\frac{ 1-\cos(x) }{ 2 })+2\sin(x) + 1 }\]

Answer 13

theres no need to use the half angle formula

Answer 14

What do I do then?

Answer 15

did you notice - my posts got swopped around weird!!

Answer 16

Sorry my internet isn't working right

Answer 17

neither is mine..

Answer 18

So after the quotient rule, what next? I don't understand how sin^2x + cos^2x can fit in because its negative in the equation.

Answer 19

ok
(-sin x - sin^2 x - cos^2 x) / (1 + sin x)^2
= - sin x - ( sin^2x + cos^2 x) / ( 1 + sin x)^2

Answer 20

now - what i s next?

Answer 21

Oh! I see. Next, do I expand the denominator? (1+sinx)^2 ?

Answer 22

no just simplify the numerator then you can cancel

Answer 23

So
(-sinx -1)/(1+sinx)^2

Answer 24

Does that become
-(1+sinx)/(1+sinx)^2
?

Answer 25

right - now you can further simplify

Answer 26

yes

Answer 27

And then
(-1)/(1+sinx)

Answer 28

right

Answer 29

I don't think I can go any further, can I?

Answer 30

no

Answer 31

Ok

Answer 32

Thank you so much for your help! I get it now.

Answer 33

yw