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Math Help - quotient rule

  1. #1
    Newbie EpicAsianDude's Avatar
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    quotient rule

    Could someone explain to me step by step of how to solve the following question using the quotient rule, much appreciated

    y= 2SIN5x/x3
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: quotient rule

    You must memorize this equation: given  y(x) = \frac {f(x)}{g(x)} then  y'(x) = \frac {f'(x)g(x) -f(x)g'(x)}{(g(x))^2} .

    For your problem you have  f(x) = 2 \sin(5x) and  g(x) = x^3. so:

     y'(x) = \frac {f'(x)g(x) -f(x)g'(x)}{(g(x))^2}  = \frac {2 \times 5 \cos(5x) x^3 - 2 \sin(5x) 3 x^2}{(x^3)^2} = \frac {10 x \cos(5x)-6 \sin(5x)}{x^4}.
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  3. #3
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    Re: quotient rule

    Hello, EpicAsianDude!~

    Could someone explain to me step by step of how to find the derivative.

    . . y \:=\: \frac{2\sin(5x)}{x^3}

    We have: . y \:=\:2\cdot\frac{\overbrace{\sin(5x)}^{f(x)}}{ \underbrace{x^3}_{g(x)}}

    Quotient Rule: . y' \;=\;2\cdot\frac{\overbrace{x^3}^{g(x)}\cdot \overbrace{5\cos(5x)}^{f'(x)} - \overbrace{\sin(5x)}^{f(x)}\cdot \overbrace{3x^2}^{g'(x)}}{\underbrace{x^6}_{[g(x)]^2}}

    . . . . . . . . . . . y' \;=\;2\cdot\frac{x^2\left[5x\cos(5x) - 3\sin(5x)\right]}{x^6}

    . . . . . . . . . . . y' \;=\;2\cdot\frac{5x\cos(5x)-3\sin(5x)}{x^4}
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  4. #4
    Newbie EpicAsianDude's Avatar
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    Re: quotient rule

    Thank you for replying ebaines & sorobans. So when answering this type of question the quotient rule formula has to be implemented in order to work out the derivative.
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  5. #5
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    Re: quotient rule

    I have another question that i need help on but relating to the chain rule, in the following question:

    y=5x^2e^2x
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  6. #6
    MHF Contributor ebaines's Avatar
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    Re: quotient rule

    Quote Originally Posted by EpicAsianDude View Post
    I have another question that i need help on but relating to the chain rule, in the following question:

    y=5x^2e^2x
    What specifically is your question? You simply apply the product rule: if  y(x) = f(x) \cdot g(x) then  y'(x) = f(x)g'(x) + f'(x) g(x). Apply this to your question and see what you get!
    Last edited by ebaines; January 27th 2014 at 10:19 AM.
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  7. #7
    Newbie EpicAsianDude's Avatar
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    Re: quotient rule

    So:

    y= 5x^2e^2x

    is equal to = 5(x^2 2e^2x + 2xe^2x)
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  8. #8
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    Re: quotient rule

    Quote Originally Posted by EpicAsianDude View Post
    So:

    y= 5x^2e^2x

    is equal to = 5(x^2 2e^2x + 2xe^2x)
    That's exactly right, and now of course you'd simplify (e.g. there are some common factors).
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