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Math Help - Choosing which answer is the best in a quotient derivative

  1. #1
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    Choosing which answer is the best in a quotient derivative

    Hello Forum, This is my first post but I would like some help. I have the equation...

    (x^4+x^4+7)/(x^4+7x^4+1)

    Now I know I got the correct answer (which is (-216x^3)/(64x^8+16x^4+1) however when I went to validate the answer on wolfram alpha.com, it gave it to me in this form.

    ((8 x^3)/(8 x^4+1))-((32 x^3 (2 x^4+7))/(8 x^4+1)^2)

    I'm sorry if its hard to understand what I'm trying to say. I don't really
    understand how to use all the little icons.

    Anyways I'd like to know which form is more appropriate. I'd assume mine would be since its more compact. However, every site I've gone to validate the answer gives it in the other form.
    My way took a very very long time to come up with and when I tried doing it in their form I got it rather easily.

    Thank you for all your help
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by firestarterut View Post
    Hello Forum, This is my first post but I would like some help. I have the equation...

    (x^4+x^4+7)/(x^4+7x^4+1)

    Now I know I got the correct answer (which is (-216x^3)/(64x^8+16x^4+1) however when I went to validate the answer on wolfram alpha.com, it gave it to me in this form.

    ((8 x^3)/(8 x^4+1))-((32 x^3 (2 x^4+7))/(8 x^4+1)^2)

    I'm sorry if its hard to understand what I'm trying to say. I don't really
    understand how to use all the little icons.

    Anyways I'd like to know which form is more appropriate. I'd assume mine would be since its more compact. However, every site I've gone to validate the answer gives it in the other form.
    My way took a very very long time to come up with and when I tried doing it in their form I got it rather easily.

    Thank you for all your help
    It's the same!

    Hint!

    64x^8+16x^4+1=(8x^4+1)^2
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    I know they result in the same answer. My question was which form is more appropriate. Notice how my form no longer has the minus sign and could no longer be reduced. The other form is still in need for a subtraction and like you stated the squared is there. So I'm wondering if people care that it can still be reduced further or is the website's form just as acceptable.
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by firestarterut View Post
    I know they result in the same answer. My question was which form is more appropriate. Notice how my form no longer has the minus sign and could no longer be reduced. The other form is still in need for a subtraction and like you stated the squared is there. So I'm wondering if people care that it can still be reduced further or is the website's form just as acceptable.
    If your answer is correct.... you can leave it in any form...
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  5. #5
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    The fact of the matter is it's all up to personal preference. Both answers are acceptable, neither is more acceptable than the other. In the context of a larger problem it might be prudent to use one over the other (as finding common factors may simplify the problem), but in this case, where you're just asked to apply a rule, either is acceptable.
    Last edited by mr fantastic; July 18th 2010 at 03:53 AM.
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    Thank you for your answer. It was appreciated.
    Last edited by mr fantastic; July 18th 2010 at 03:53 AM.
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    Quote Originally Posted by firestarterut View Post
    Hello Forum, This is my first post but I would like some help. I have the equation...

    (x^4+x^4+7)/(x^4+7x^4+1)

    Now I know I got the correct answer (which is (-216x^3)/(64x^8+16x^4+1) however when I went to validate the answer on wolfram alpha.com, it gave it to me in this form.

    ((8 x^3)/(8 x^4+1))-((32 x^3 (2 x^4+7))/(8 x^4+1)^2)

    I'm sorry if its hard to understand what I'm trying to say. I don't really
    understand how to use all the little icons.

    Anyways I'd like to know which form is more appropriate. I'd assume mine would be since its more compact. However, every site I've gone to validate the answer gives it in the other form.
    My way took a very very long time to come up with and when I tried doing it in their form I got it rather easily.

    Thank you for all your help
    Hi firestarterut,

    applying the quotient rule of differentiation..

    u=2x^4+7\ \Rightarrow\ \frac{du}{dx}=8x^3

    v=8x^4+1\ \Rightarrow\ \frac{dv}{dx}=32x^3

    then applying the formula

    \frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}

    gives

    \frac{\left(8x^4+1\right)8x^3-\left(2x^4+7\right)32x^3}{\left(8x^4+1\right)^2}

    Now you can split this into

    \frac{\left(8x^4+1\right)8x^3}{\left(8x^4+1\right)  \left(8x^4+1\right)}-\frac{\left(2x^4+7\right)32x^3}{\left(8x^4+1\right  )^2}

    and simplify as it was done online,
    or you can simplify using your own method as you noticed the x^7 terms cancel in the numerator.

    \frac{8x^3-(7)32x^3}{\left(8x^4+1\right)^2}

    Your final form is more compact,
    however, if a value of x is placed into either form,
    then the evaluation of the derivative of the quotient will be the same for any x.

    You were able to solve it and that's what matters.
    Solutions in terms of the variable can be left in a number of forms all equal.
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  8. #8
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    Quote Originally Posted by firestarterut View Post
    Hello Forum, This is my first post but I would like some help. I have the equation...

    (x^4+x^4+7)/(x^4+7x^4+1)

    Now I know I got the correct answer (which is (-216x^3)/(64x^8+16x^4+1) however when I went to validate the answer on wolfram alpha.com, it gave it to me in this form.

    ((8 x^3)/(8 x^4+1))-((32 x^3 (2 x^4+7))/(8 x^4+1)^2)

    I'm sorry if its hard to understand what I'm trying to say. I don't really
    understand how to use all the little icons.

    Anyways I'd like to know which form is more appropriate. I'd assume mine would be since its more compact. However, every site I've gone to validate the answer gives it in the other form.
    My way took a very very long time to come up with and when I tried doing it in their form I got it rather easily.

    Thank you for all your help
    The fact is that both answers are correct and that's it. Unless the question specifies the form that the final answer must be in, any correct answer is valid.

    If you want to know which is more acceptable, I suggest you tell us to whom you want to know which is more acceptable. Then I suggest you ask THAT person which is more acceptabe to them.

    Thread closed.
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