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Multivariable Chain Rule of sin(x)cos(2y)

  1. Oct 18, 2012 #1
    Hi all, I've got a Calculus III Question

    1. The problem statement, all variables and given/known data
    Find the derivative zs and zt, where z=sin(x)cos(2y)


    2. Relevant equations
    x=s+t
    y=s-t

    3. The attempt at a solution

    I had a go at the solution and this was what I ended up getting

    for zs, I ended up getting (cosxcos2y)(1)-2sinxsin2y(1)

    where the 1 at the end of the δx and δy were the partial derivatives of x=s+t and y=s-t

    I subbed in s and t values for x and y, and I ended up with

    zs=((cos(s)+cos(t))(cos(2s)-cos(2t)))-((2sin(s)+2sin(t))(sin(2s)-sin(2t)))

    I did the same process for t and got

    zt=((-cos(s)-cos(t))(-cos(2s)+cos(2t)))+((2sin(s)+2sin(t))(sin(2s)-sin(2t)))

    this seems like a really unnecessarily long answer and i'm pretty sure I messed something up. I can't seem to find the mistake though (I have a feeling its right under my nose).

    Also, I don't know if this is really the easiest way to go about doing these problems. I felt like the past few chapters we've learned in lecture have been kinda rushed. If there is any way where these problems could be solved in an easier way, that advice would be MUCH appreciated. Thanks

    --my brain feels like its turning into mush

    P.S. if there are any post-editing mistakes, please forgive me. I am not used to posting on this site yet.
     
    Last edited: Oct 18, 2012
  2. jcsd
  3. Oct 18, 2012 #2

    SammyS

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    Hello SweetBabyLou. Welcome to PF !

    You're correct up to this point.

    What you have next, is incorrect.
    cos(a+b) ≠ cos(a) + cos(b), etc.

    Use angle addition identities.
     
  4. Oct 19, 2012 #3
    Hi SammyS,

    I'm sorry, but I'm not really familiar with Angle Addition Identities (It is probably something I've learned, but has slipped my mind). I did, however, look it up on the friendly neighborhood Google, and saw in Wolfram Alpha's MathWorld, that if I had sin(a+b) (or in this case, s+t) then the result should be something along the lines of, sin(a)cos(b)+sin(b)cos(a). I also see that for cos(a+b) the result should be cos(a)cos(b)-sin(a)sin(b). Am I headed in the right direction?
     
  5. Oct 19, 2012 #4

    SammyS

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    Yes, those are the angle addition identities .
     
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