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Problems with implicit differentiation

  1. May 10, 2007 #1
    In a problem where I need to use implicit diff. to find the slope of a line such as:

    y^2 + x^2 = 9

    2y (dy/dx) + 2x = 0

    dy/dx = -y/x

    Where does the (dy/dx) after the 2y (in the second part) come from? I've already differentiated y^2, x^2, and 9. Why isn't it just 2y + 2x = 0? I've looked all over, and in every problem an extra (dy/dx) or two just seem to pop up out of nowhere. Thanks!
  2. jcsd
  3. May 10, 2007 #2
    Think of Y as a function in X. Then it's just the chain rule.

    For example if y=e^x+x^3
    y' = e^x + 3x^2

    y^2 + x^2 = 9 is

    (e^x+x^3)^2 + x^2 - 9 = 0
    when we plug in y

    differentiating this we get
    2(e^x+x^3)(e^x + 3x^2) + 2x = 0

    or 2y*(dy/dx) +2x = 0

    when you do implicit differentiation you are doing it for a general y function of x instead of something specific (like as in our case of y=e^x+x^3)
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