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Math Help - Tricky (for me anyway) Differentiation.

  1. #1
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    Tricky (for me anyway) Differentiation.

    Hi

    I am trying to differentiate.
    Many attempts at trying, and I cannot get to the answer. I am doing something wrong. Just having the answer is not helping. I have tried using the product and quotient rule(s) but am probably not simplifying correctly. If someone can show the steps that would really be great.

    \frac{x^2-1}{\sqrt{x^2+1}}

    Thanks
    Regards
    Craig.
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  2. #2
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    Quote Originally Posted by craigmain View Post
    Hi

    I am trying to differentiate.
    Many attempts at trying, and I cannot get to the answer. I am doing something wrong. Just having the answer is not helping. I have tried using the product and quotient rule(s) but am probably not simplifying correctly. If someone can show the steps that would really be great.

    \frac{x^2-1}{\sqrt{x^2+1}}

    Thanks
    Regards
    Craig.
    Use the quotient rule. To get the derivative of the denominator, use the chain rule.

    If you post your work it will be easier to know where you're going wrong.
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  3. #3
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    Tricky (for me anyway) Differentiation

    Here are some steps.

    Here is the start.
    <br />
\frac{2x.\sqrt{x^2+1}+\frac{x.\sqrt{x^2+1}}{(x^2+1  )^2}.(x^2-1)}{(x^2+1)}<br />

    Then
    <br />
\frac{(x.\sqrt{x^2+1}).(2(x^2+1)^2+(x^2-1))}{(x^2+1)^3}<br />

    But something is wrong. I need to get to:

    <br />
\frac{x.(x^2+3).\sqrt{x^2+1}}{(x^2+1)^2}<br />

    I cannot see how simplifying will get the terms I need to cancel.

    Cannot seem to get there. I have pages of working, and I am doing something stupid.
    Last edited by craigmain; February 8th 2009 at 03:30 AM.
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  4. #4
    Moo
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    Hello,
    Quote Originally Posted by craigmain View Post
    Here are some steps.

    Here is the start.
    <br />
\frac{2x.\sqrt{x^2+1}+\frac{x.\sqrt{x^2+1}}{(x^2+1  )^2}.(x^2-1)}{(x^2+1)}<br />

    Then
    <br />
\frac{(x.\sqrt{x^2+1}).(2(x^2+1)^2+(x^2-1))}{(x^2+1)^3}<br />

    But something is wrong. I need to get to:

    <br />
\frac{x.(x^2+3).\sqrt{x^2+1}}{(x^2+1)^2}<br />

    I cannot see how simplifying will get the terms I need to cancel.

    Cannot seem to get there. I have pages of working, and I am doing something stupid.
    You went wrong in the differentiation (1st step)

    Quotient rule says :
    \left[\frac uv\right]'=\frac{u'v-uv'}{v^2}

    But what you did is \frac{u'v-u \left(\frac 1v\right)'}{v^2}

    Because -\frac{x\sqrt{x^2+1}}{(x^2+1)^2} is the derivative of \frac{1}{\sqrt{x^2+1}}, not the derivative of \sqrt{x^2+1}
    Just a careless mistake, I guess ^^'
    Last edited by Moo; February 8th 2009 at 04:07 AM. Reason: forgot a square
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  5. #5
    Like a stone-audioslave ADARSH's Avatar
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    Dont try to jump over the steps
    I'll give you an example to differentiate

     <br />
\frac{x-1}{\sqrt{x}}<br />
    After using quotient rule the differntiation becomes

     <br /> <br />
\frac{\sqrt{x} \frac{d(x-1)}{dx} - (x-1)\frac{d(\sqrt{x})}{dx} }{\sqrt{x}^2}<br /> <br />

    Now try using it in your case and watch the use of quotient rule
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  6. #6
    Moo
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    Quote Originally Posted by ADARSH View Post

    Dont try to jump over the steps
    I'll give you an example to differentiate

     <br />
\frac{x-1}{\sqrt{x}}<br />
    After using quotient rule the differntiation becomes

     <br /> <br />
\frac{\sqrt{x} \frac{d(x-1)}{dx} - (x-1)\frac{d(\sqrt{x})}{dx} }{\sqrt{x}^2}<br /> <br />

    Now try using it in your case and watch the use of quotient rule
    He indeed jumped some steps, but he did everything perfectly (or maybe he did more on his paper)
    It's just that he didn't apply the formula correctly :P
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  7. #7
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    Quote Originally Posted by craigmain View Post
    Here are some steps.

    Here is the start.
    <br />
\frac{2x.\sqrt{x^2+1}+\frac{x.\sqrt{x^2+1}}{(x^2+1  )^2}.(x^2-1)}{(x^2+1)}<br />
    Maybe you are correct (And I I know it will be the case)
    But
    I don't think differentiation of denominator in first case is correct
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  8. #8
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    You might find it simpler to write you function as (x^2- 1)(x^2+ 1)^{-\frac{1}{2}} and use the product rule.
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  9. #9
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    f(x)=\frac{{{x}^{2}}-1}{\sqrt{{{x}^{2}}+1}}=\frac{{{x}^{2}}+1-2}{\sqrt{{{x}^{2}}+1}}=\sqrt{{{x}^{2}}+1}-2{{\left( {{x}^{2}}+1 \right)}^{-\frac{1}{2}}}, so

    f'(x)=\frac{x}{\sqrt{{{x}^{2}}+1}}+\frac{2x}{\sqrt  {{{\left( {{x}^{2}}+1 \right)}^{3}}}}.
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