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Math Help - Difference quotient!

  1. #1
    Member rowe's Avatar
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    Difference quotient!

    Need to show that \frac{\frac{1}{x}-\frac{1}{a}}{x-a} = -\frac{1}{ax}.

    Basically, I'll do something like \left(\frac{1}{x}-\frac{1}{a}\right)\frac{1}{x-a}, and manipulate it in several different ways, all of which are far more complicated than the simplified form, like \frac{a-x}{ax^2-a^2x}. What's the trick here?
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  2. #2
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by rowe View Post
    Need to show that \frac{\frac{1}{x}-\frac{1}{a}}{x-a} = -\frac{1}{ax}.

    Basically, I'll do something like \left(\frac{1}{x}-\frac{1}{a}\right)\frac{1}{x-a}, and manipulate it in several different ways, all of which are far more complicated than the simplified form, like \frac{a-x}{ax^2-a^2x}. What's the trick here?
    Note that ax^2-a^2x = ax(x-a) and (x-a) = -(a-x)
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  3. #3
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    Quote Originally Posted by rowe View Post
    Need to show that \frac{\frac{1}{x}-\frac{1}{a}}{x-a} = -\frac{1}{ax}.

    Basically, I'll do something like \left(\frac{1}{x}-\frac{1}{a}\right)\frac{1}{x-a}, and manipulate it in several different ways, all of which are far more complicated than the simplified form, like \frac{a-x}{ax^2-a^2x}. What's the trick here?
    Actually, the first thing I would have done is get rid of the "compound" fraction by multiplying numerator and denominator on the left by ax:
    \frac{ax(\frac{1}{x}- \frac{1}{a})}{ax(x- a)} which immediately gives your \frac{a-x}{ax(x-a)} and then cancel:
    \frac{-(x-a)}{ax(x-a)}= -\frac{1}{ax}
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