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How to differentiate 2x/(x+y) = y, implicitly?

  1. Apr 23, 2005 #1
    differentiate
    2x/(x+y)=y

    Method1:
    2x=xy+y^2
    d/dx(2x)=d/dx(xy+y^2)
    2=y+xy'+2y(y')
    y'=(2-y)/(x+2y)

    Method2:
    2x/(x + y) = y
    y' = [2(x + y) - (2x)(1 + y')]/(x + y)^2
    [(x + y)^2 + 2x]y' = 2(x + y) - 2x
    y' = 2y/[(x + y)^2 + 2x]

    Which method is using "implicit" differentiation?
     
  2. jcsd
  3. Apr 23, 2005 #2
    Both are, but the second is alot sloppier.
     
  4. Apr 23, 2005 #3
    another question:
    x^3+y^3=6xy

    3x^2+(3y^3)y'=...?
    how do u differentiate 6xy?
     
  5. Apr 23, 2005 #4
    product rule
     
  6. Apr 23, 2005 #5

    Zurtex

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    Product rule! :wink:
     
  7. Apr 23, 2005 #6
    i only learnt how to differentiate using prodct rule with two terms....is it the same with 3 terms?
     
  8. Apr 23, 2005 #7
    6 is a constant, dont even worry about it.
     
  9. Apr 23, 2005 #8
    so....(6xy')+(6y)....?
     
  10. Apr 23, 2005 #9
    6xy

    u(x) = 6x, v(x) = y

    (u(x)v(x))' = u'(x)v(x) + v'(x)u(x)
     
  11. Apr 23, 2005 #10
    did i start this right?
    x^2+y^2+2x-4y-20=0
    2x+(2y)(y')+2+4y'=0
     
  12. Apr 23, 2005 #11
    you seem to have made a sign error, but other than that it's okay.
     
  13. Apr 23, 2005 #12
    o..i see it...thanks
     
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