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Implicit Differentiation

  1. Nov 1, 2009 #1
    1. The problem statement, all variables and given/known data
    Find all the points on the curve [tex]x^{2}y^{2}+xy=2[/tex] where the slope of the tangent line is -1.



    3. The attempt at a solution
    I differentiated both sides of the equation and got:
    [tex]\frac{dy}{dx}=\frac{-2xy^{2}-y}{x^{2}2y+x}[/tex]

    I know that [tex]\frac{dy}{dx}=-1[/tex], but if I substitute -1 in, I won't be able to go any further since I have two unknown variables. I would appreciate any help.
     
    Last edited: Nov 1, 2009
  2. jcsd
  3. Nov 1, 2009 #2

    Mark44

    Staff: Mentor

    [tex]\frac{dy}{dx}=\frac{-2xy^{2}-y}{2x^{2}y+x}~=~\frac{-y(2xy + 1)}{x(2xy + 1)}[/tex]

    As long as 2xy + 1 [itex]\neq[/itex] 0, you can cancel the factors of 2xy + 1, leaving a much simpler derivative.

    Also, you want to solve the equation dy/dx = -1, not dy/dx = 1, as you had. Notice that you still have two variables, but all that means is that there are lots of solutions.
     
  4. Nov 1, 2009 #3
    Thanks, I got it.
     
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