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Math Help - Algebra Help for a Derivative

  1. #1
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    Algebra Help for a Derivative

    There's probably easier ways to solve this derivative by using logarithmic rules, but using the method that I took, do you see where I go wrong with the algebra?

    It would greatly help to see what errors I making! I think the same errors pop up when doing problems like this. Thank you!
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  2. #2
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    Re: Algebra Help for a Derivative

    I can't really read what you wrote. It looks like the problem is y = \ln\left( \dfrac{(x^2+1)^5}{\sqrt{2-x}} \right). Then, when you get to line 6, it appears that \dfrac{1}{\left(\dfrac{(x^2+1)^5}{\sqrt{2-x}}\right) } becomes \dfrac{1}{\sqrt{2-x} (x^2+1)^5}\right) }. That is wrong. It should become \dfrac{\sqrt{2-x}}{(x^2+1)^5}
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    Re: Algebra Help for a Derivative

    Yes! That's what puzzles me a lot. I don't have that part down in my head yet.

    When dividing a complex fraction like that, I confuse the order of what group divides what.

    Any advice on that? I took it that 1 divides (x^2 +1)^5 first.
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    Re: Algebra Help for a Derivative

    Multiply numerator and denominator by \sqrt{2-x}. You get 1\cdot \sqrt{2-x} in the numerator and \dfrac{(x^2+1)^5}{\sqrt{2-x}} \sqrt{2-x} in the denominator. The \sqrt{2-x} and the \dfrac{1}{\sqrt{2-x}} cancel each other out.
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    Re: Algebra Help for a Derivative

    Algebra Help for a Derivative-07-oct-13.png
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    Re: Algebra Help for a Derivative

    Looks good with a quick glance
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    Re: Algebra Help for a Derivative

    Quote Originally Posted by IdentityProblem View Post
    There's probably easier ways to solve this derivative by using logarithmic rules, but using the method that I took, do you see where I go wrong with the algebra?

    It would greatly help to see what errors I making! I think the same errors pop up when doing problems like this. Thank you!
    Why don't you use the logarithm laws to make life easier on yourself?

    \displaystyle \begin{align*} \ln{ \left[ \frac{ \left( x^2 + 1 \right) ^5 }{\sqrt{1 - x} } \right] } &= \ln{ \left[ \left( x^2 + 1 \right) ^5 \right] } - \ln{ \left( \sqrt{1 - x} \right) } \\ &= 5\ln{ \left( x^2 + 1 \right) } - \ln{ \left[ \left( 1 - x \right) ^{\frac{1}{2}} \right] } \\ &= 5 \ln{ \left( x^2 + 1 \right) } - \frac{1}{2} \ln{ \left( 1 - x \right) }  \end{align*}

    These functions are MUCH easier to differentiate...
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