Expert: Paul Klarreich - 2/8/2006

Question
x^2+6xy+y^2=8

please show me your work


Answer
Hi, nic,
You wrote:
Subject:  I cant find the second derivative please help
Question:  x^2+6xy+y^2=8

please show me your work
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First you find the derivative, using implicit differentiation: (I don't like to use the y' notation, but the dy/dx notation would be really bad here, so...)

WARNING: THE MATERIAL BELOW MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  BE SURE TO VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.

2x + 6y + 6x y' + 2y y' = 0

x + 3y + 3x y' + y y' = 0

          (3x + y) y' = - x - 3y

    -x - 3y
y' = -------
     3x + y

OK. Now here's the scheme:  

1. Use the quotient rule to differentiate that.   Since y still appears in it, y' will appear in the result.

2. Where you see y', substitute the result we just got.

2.5:  Prepare a pot of black coffee and find some aspirin.  Ready?

3. Simplify.

     (3x + y )(-1 - 3y' ) - (-x - 3y)(3 + y')
y'' = ----------------------------------------
               (3x + y)^2  


     (3x + y )(-1 - 3y' ) + (x + 3y)(3 + y')
y'' = ----------------------------------------
               (3x + y)^2  


     (3x + y )(-1) - (3x + y) 3y' + (x + 3y)3 + (x + 3y)y'
y'' = -----------------------------------------------------
                 (3x + y)^2  


     -3x  - y  - 9xy' - 3yy'  + 3x  + 9y + xy' + 3yy'
y'' = ------------------------------------------------
                 (3x + y)^2  


          - y  - 9xy'               + 9y + xy'
y'' = ------------------------------------------------
                 (3x + y)^2  


     8y  - 8xy       y - x[(-x-3y)/(3x+y)]
y'' = ----------- = 8 ---------------------
      (3x + y)^2            (...)^2

Simplify by multiplying each term by  (3x+y):
       y(3x + y) - x(-x - 3y)
y'' = 8 ----------------------
           (...)^3

       3xy + y^2 + x^2 + 3xy
y'' = 8 ---------------------
           (...)^3


       x^2 + 6xy + y^2
y'' = 8 ----------------
           (3x + y)^3


Hope I didn't blow too many signs in that.
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