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Definition of Sec

  1. Mar 7, 2012 #1
    1. The problem statement, all variables and given/known data
    Having started to differentiate Ln(cos(5x)) wrt x I checked the answer with WolframAlpha & got a different method & answer too.


    2. Relevant equations
    differentiate
    ln(cos(5x))



    3. The attempt at a solution
    I used the Chain rule d(ln(cos(5x)))/dx = d(ln(u))/du * du/dx
    Where u = cos(5x)

    d(ln(u))/du = 1/u which I have in a table of standard derivatives
    = 1/(cos(5x)

    du/dx = -sin(5x) again, from a table of standard derivatives.

    So according to me d(ln(cos(5x)))/dx = 1/cos(5x) * -sin(5x)


    WolframAlpha goes a different route & has the answer
    -5 tan(5x)

    I see how this differs from mine so;
    1. why am I wrong,
    2. how did WA get from sec(5x)(sin(5x)(-(d(5x)/dx))) to -5tan(5x). i.e how is tan defined in terms of sec?
    Sorry to be dense!
     
    Last edited: Mar 7, 2012
  2. jcsd
  3. Mar 7, 2012 #2
    Something wrong here~

    and sec x = 1 / cos x
     
  4. Mar 7, 2012 #3
    Hmmm!
    My table of standard derivatives shows;
    if function f(x) = cos(ax),
    the derivative f'(x) = -a sin(ax)

    In my example a = 5, so -5 sin(5x)

    Ah! I see the bit I missed out.
     
    Last edited: Mar 7, 2012
  5. Mar 7, 2012 #4

    Ray Vickson

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    Homework Helper

    What is the *definition* of tan(w)? Surely your textbook tells you that! If not, try Google.

    RGV
     
  6. Mar 7, 2012 #5
    Er! Tan (w)= opposite/adjacent

    tan(w) = sin(w)/cos(w)
    sec(w) = 1/cos(w)

    I haven't yet found a relationship between sec (w) & tan(w) which also involves multiplying it by sin(w).
    From the WA page I can, of course, see that it has the relationship as tan(w) = sec(w)sin(w). I would like someone to confirm this, please & say whether this is a standard which I haven't yet encountered.
     
    Last edited: Mar 7, 2012
  7. Mar 7, 2012 #6

    HallsofIvy

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    Staff Emeritus
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    You have it right there- you originally gave, as the derivative, (1/cos(5x))(-sin(5x)). You now recognize that, because the derivative of 5x is 5, it should be (5)(1/cos(5x)(-sin(5x))= -5(sin(5x)/cos(5x))= -5tan(5x).
     
  8. Mar 8, 2012 #7
    Amazingly easy when pointed out.
    . . . 'wood for the trees - again!

    Completed my original question
    f(x)=sec(5x)ln(cos(5x))
    after product rule & then chain rule for different bits.
    Thank you all.
     
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