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Math Help - Why getting different results when using product rule over quotient rule?

  1. #1
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    Why getting different results when using product rule over quotient rule?

    Hello

    I have a function:
    y = \frac{e^x}{x}

    If I differentiate using the quotient rule, then:
    let u = e^x and let v = x

    \frac{dy}{dx} = \frac{x.e^x - e^x.1}{x^2}

    = \frac{e^x(x - 1)}{x^2}

    But could I not express as a product ie y = e^x . x^-1

    And then use the product rule,
    let u = e^x and v = x^-1

    \frac{dy}{dx} = x^-1 . e^x - e^x . x^-2

    \frac{e^x}{x} - \frac{e^x}{x^2} = \frac{e^(2x)}{x^2} - \frac{e^x}{x^2}

    = \frac{e^(2x) - e^x}{x^2}

    Which is a different result! Why so? Or have I done it wrong?
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  2. #2
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    Re: Why getting different results when using product rule over quotient rule?

    You made a mistake on the second last line.

    \frac{e^x}{x} - \frac{e^x}{x^2} = \frac{xe^x}{x^2} - \frac{e^x}{x^2} = \frac{xe^x-e^x}{x^2}
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  3. #3
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    Re: Why getting different results when using product rule over quotient rule?

    Quote Originally Posted by SpringFan25 View Post
    You made a mistake on the second last line.

    \frac{e^x}{x} - \frac{e^x}{x^2} = \frac{xe^x}{x^2} - \frac{e^x}{x^2} = \frac{xe^x-e^x}{x^2}
    Ah I squared top and bottom rather than multiply by x. I assumed squaring top and bottom would work.

    I worked it out with a real example:

    \frac{3}{4} - \frac{1}{16}

    incorrect square method:
    \frac{9}{16} - \frac{1}{16} =  \frac{8}{16} =  \frac{1}{2} WRONG!


    proper way - multiply top and bottom by 4
    \frac{12}{16} - \frac{1}{16} =  \frac{11}{16} or (0.6875) CORRECT!

    Well, I have learnt something. Thank you.
    Last edited by angypangy; December 20th 2011 at 02:24 AM.
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  4. #4
    Super Member Quacky's Avatar
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    Re: Why getting different results when using product rule over quotient rule?

    Consider the case of \frac{4}{9}. Is this equal to:

    \frac{4}{9}\times\frac{9}{9}

    ...or \frac{4}{9}\times\frac{4}{9}?

    You have to multiply the numerator and the denominator by the same value in order for them to be equivalent.
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  5. #5
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    Re: Why getting different results when using product rule over quotient rule?

    Quote Originally Posted by Quacky View Post
    Consider the case of \frac{4}{9}. Is this equal to:

    \frac{4}{9}\times\frac{9}{9}

    ...or \frac{4}{9}\times\frac{4}{9}?

    You have to multiply the numerator and the denominator by the same value in order for them to be equivalent.
    I answered my own question in the end. Yes agree (now) - thanks. I am learning by mistake
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  6. #6
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    Re: Why getting different results when using product rule over quotient rule?

    Quote Originally Posted by angypangy View Post
    I answered my own question in the end. Yes agree (now) - thanks. I am learning by mistake
    Mistakes are the gems of learning. Whenever you make a mistake, assuming the stakes are low such as in a classroom and not as in building a bridge, rejoice that you found it! Then hammer away at that mistake until you know exactly what you did wrong, and have come up with a method for avoiding that mistake in the future.
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