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Having a hard time with a derivative

  1. Apr 27, 2005 #1
    Can not seem to find the answer. If x^sin y= y^cos x

    Find dx/dy (pi/4 , pi/4).
    If someone could help it would be great.
     
  2. jcsd
  3. Apr 27, 2005 #2

    Pyrrhus

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    Why don't you use logarithms properties to do it?
     
  4. Apr 27, 2005 #3
    I've tried and came out with x'=-1.07
    Can you check this answer.
     
  5. Apr 28, 2005 #4
    wat does trhe value iun the bracket signify if i know that i could help
     
  6. Apr 28, 2005 #5
    the values are (3.14/4, 3.14/4) or (pie/4, pie/4)
     
  7. Apr 28, 2005 #6

    saltydog

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    Rolando, I get a different answer. Did you get:

    [tex]\frac{dx}{dy}=\frac{\frac{Cos(x)}{y}-Ln(x)Cos(y)}{\frac{Sin(y)}{x}+Ln(y)Sin(x)}[/tex]

    When I plug in x=pi/4 and y=pi/4 I get 1.4683
     
  8. Apr 29, 2005 #7
    [tex]x^{sin(y)} = y^{cos(x)}[/tex]

    is equivalent to

    [tex] \frac{ln(x)}{cos(x)} = \frac{ln(y)}{sin(y)}[/tex]

    Implicitly differentiating, I got

    [tex]\frac{\dy}{\dx} = \frac {sin^{2}(y) \left( \frac{cos(x)}{x} + sin(x)ln(x) \right) } {cos^{2}(x) \left( \frac{sin(y)}{y} - cos(y)ln(y) \right) } [/tex]

    which, at (pi/4, pi/4), is

    [tex]= \frac{ \frac{4}{\pi} + ln \left( \frac{\pi}{4} \right) }{\frac{4}{\pi} - ln \left( \frac{\pi}{4} \right)}[/tex]

    I don't have a calculator to approximate, though.
     
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