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Differentiation anomoly-in need of setting straight!

  1. May 7, 2007 #1
    Differentiation anomoly--in need of setting straight!

    I was studying for the AP calculus exam when I came accross this question, which seems to have multiple answers. I cannot seem to find what is wrong with it (nor can my calc teacher).

    Find dy/dx if x + y = x * y

    I went about this problem the "easy way out". Instead of implicitly differentiating, I figured I could do some algebra to simplify it to y = x / (x-1). I saw the answers, and only one of them did not have a "y" in the answer, so I picked that one. It was wrong.

    When solved implicitly, a different answer is found: (y-1)/(1-x). This is STILL different than what our answer key has, which is (1-y)/(x-1).

    Can someone please help resolve this? Thanks!
  2. jcsd
  3. May 7, 2007 #2

    matt grime

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    What ever other problems you're having:

    contains a mistake on your part. The two answers written there are the same.
  4. May 7, 2007 #3
    If you differentiate [tex]y=\frac{x}{x-1}[/tex], you get [tex]\frac{dy}{dx}=\frac{(x-1)-x}{(x-1)^2}[/tex].

    Divide by [tex]\frac{x-1}{x-1}: \frac{dy}{dx}=\frac{1-\frac{x}{x-1}}{x-1}[/tex]

    Substitute [tex]y=\frac{x}{x-1}[/tex]: [tex]\frac{dy}{dx}=\frac{1-y}{x-1}[/tex]
  5. May 7, 2007 #4


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    No.... it's just negative on the top and bottom. Multiply your answer by 1 = -1/-1 and see what you get
  6. May 7, 2007 #5


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    Isn't implicit differentiation "the easy way"?

    1 + dy/dx = y + x dy/dx

    1-y = (x-1)dy/dx

    dy/dx = (1-y)/(x-1)
  7. May 7, 2007 #6
    What I meant is that, because I could see that it simplified to y = *stuff with no y*, I figured that dy/dx MUST not have a y in it. There was only one choice without one, so I chose that. It was the easy way out because I did not do any differentiating at all.

    Also, I figured out my problem. Both answers were the same, but had a "y", and the other did not. When a substitution was made for "y", the answers were identical.

    And yeah... I can't believe I didn't notice that the answer simply divided by a factor of (-1). Wow I feel dumb! :-p

    All here is well, thank you for the help.
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