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Find f prime (x) if f(x) = (cosx)/(1+secx)

Mathematics
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have you tried quotient rule?
Yes, I did and then I don't know how to distribute? I know I have to use it It just doesn't fully come out
Show us how far you got with the quotient rule. Show all the steps you have so far. Can you do that?

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Other answers:

Yes, I get (|dw:1350172024457:dw|
Correct! So far! Now what do you think the next step should be to simplify?
You're only going to simplify the numerator. Never expand the denominator. Do you understand?
Yes. Should I distribute the -sinx to (1+secx)?
Go ahead and apply the distributive property to the numerator. In other words, expand.
Yes! Go ahead. Show me the next step.
|dw:1350172306983:dw| I shouldn't distribute the cosx correct? Because tanx and secx are being multiplied together?
Yes expand the entire numerator. Meaning I shouldn't see any more parentheses in the numerator. Go ahead!
After that change each sec x to 1/cos x. Understood? Go ahead.
I mean change each sec x to 1/cos x only in the numerator. Never expand the denominator, like I said earlier. Do you understand?
Yes. I am just confused on what I should do to the -cosx(tanxsecx)
\[-\cos x(\tan x \sec x)=-(\cos x)(\tan x)(\frac{ 1 }{ \cos x })\]Agreed?
So do exactly that. Go ahead.
Now show me the step in it's entirety.
Okay.
|dw:1350172978272:dw|
\[f \prime(x)=\frac{ -\sin x(1+\sec x)-\cos x(\sec x \tan x) }{(1+\sec x)^{2} }\] \[f \prime(x)=\frac{ -\sin x -\sin x \sec x -\cos x \sec x \tan x }{ (1+\sec x)^{2} }\] \[f \prime(x)=\frac{ -\sin x -\sin x(\frac{ 1 }{ \cos x })-\cos x(\frac{ 1 }{ \cos x })\tan x }{ (1+\sec x)^{2} }\] Take a moment to look at this the first couple of steps are exactly what you did. Do you understand what I did in the third step and now first tell me what you think you should do.
You are correct in that you should change the sin x/cosx to tan x. Go ahead and show me the next step.
Can you multiply sinx and put it over cosx to get tangent on the left too?
Of course!\[\sin x(\frac{ 1 }{ \cos x })=\frac{ \sin x }{ \cos x }=\tan x\]
|dw:1350174177943:dw|
Have more confidence in yourself, would ya! You're doing just fine. Now continue with confidence. And even if you're wrong, so what? Being wrong is part of learning and part of life. So never be afraid to ask questions even if you think you're wrong. Now could you please show me the remaining steps, with confidence!
Not quite. There is an error in the numerator but it just seems to be a careless one. Retrace your steps to see if you can spot the error. Otherwise show me your previous step so that I can point it out to you.
|dw:1350174369510:dw| Thank you for all your help
No! Now look back carefully at the third step of the partial solution, I posted above. Is sin x multiplied by tan x? I don't think so.
The simplified answer should be\[f \prime(x)=\frac{ -\sin x -2\tan x }{ (1+\sec x)^{2} }\]
|dw:1350174836525:dw|
So you understand where you went wrong?
Do you understand how to do this now?
Yes, I think so, its just such a long process that I feel like I make mistakes so easily.
Thank you so much for your help, thanks for following the process with me!
By the way, if the question doesn't ask you to simplify then it is not wrong to even leave your derivative unsimplified. However every teacher is different so you may want to consult with yours. I prefer students to simplify to a certain degree unless I specifically ask not to simplify. But as I said every teacher is different.
My pleasure. If you'd like any more help, let me know.
Thank you so much!!
Welcome anytime!

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