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Implicit Differentiation

  1. Oct 20, 2009 #1
    Find [tex]d^2/dx^2(3y^2+8y=3x)[/tex]

    I managed to get [tex]dy/dx = 3 / (6y + 8)[/tex] but I have no clue where to go from here.

    According to WolfRamAlpha, the answer is [tex]-27/(4(16 + 9x)(4 + 3y))[/tex], but since dy/dx doesn't have any x value in it, I don't see how the derivative of it would.

    I've played around with it for a long time, and I just can't get it. Help please?
     
  2. jcsd
  3. Oct 20, 2009 #2
    Do you need to express in terms of x and y? I don't see whats wrong if you express it in y which is what i would do
     
  4. Oct 20, 2009 #3
    You can't express y in terms of x. It's not a function. You can express it in terms of x and y, but that just makes the math longer.
     
  5. Oct 20, 2009 #4
    Duncan - WolfRamAlpha mad a substitution in the denominator using the original expression.

    Firstly - did you compute the second derivative correctly? You should have yielded:

    [tex]
    y' =\frac{3}{8+6 \cdot y}
    [/tex]
    and
    [tex]
    y'' = -\frac{6 \cdot (y')^2}{8+6 \cdot y}
    [/tex]

    Now, using the first into the second

    [tex]
    y'' = -\left (\frac{3}{8+6 \cdot y} \right)^2 \left (\frac{6}{8+6 \cdot y} \right ) = \frac{-54}{8 \cdot (4+3 \cdot y)^3}
    [/tex]

    Now, using the fact that [itex]3y^2+8y=3x[/itex], you need to show yourself that [itex](4+3y)^2 = 9x +16[/itex]. Use this in above to yield result.
     
  6. Oct 20, 2009 #5
    Oh. Now I get it.

    Thank you.
     
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