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Math Help - Derivative problem

  1. #1
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    Derivative problem

    sqrt((cot^2(x))+x^2)

    My attempt-

    sqrt(2cot(x) * (-csc^2x) +2x)<br />
    Here I believe I have two options.. not sure which one to go with if either..
    sqrt(2cot(x) * (-cot^2x - 1) +2x)<br />
    or

    sqrt(2cot(x) * (1/sin^2x) +2x) = sqrt(2cosx/sinx * (1/sin^2x)+2x)

    just trying to get it to simplest terms.
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  2. #2
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    do you know how the chain rule works?

    \frac{d}{dx} \left[(\cot^2{x} + x^2)^{\frac{1}{2}}\right]

    \frac{1}{2}(\cot^2{x} + x^2)^{-\frac{1}{2}} \cdot (-2\cot{x}\csc^2{x} + 2x)

    I'll leave the very little bit of possible simplification to you.
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  3. #3
    Member javax's Avatar
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     <br />
(\sqrt{\cot^2 x+x^2}) = \frac{(\cot^2 x)'+(x^2)'}{2\sqrt{\cot^2 x+x^2}} = \frac{-2\cot x \csc^2 x + 2x}{2\sqrt{\cot^2 x+x^2}} = \frac{x-\cot x \csc^2 x}{\sqrt{\cot^2 x+x^2}}<br />

    I don't think it can get any simplier.
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  4. #4
    Member javax's Avatar
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    Quote Originally Posted by skeeter View Post
    do you know how the chain rule works?

    \frac{d}{dx} \left[(\cot^2{x} + x^2)^{\frac{1}{2}}\right]

    \frac{1}{2}(\cot^2{x} + x^2)^{-\frac{1}{2}} \cdot (-2\cot{x}\csc^2{x} + 2x)

    I'll leave the very little bit of possible simplification to you.
    sorry I didn't see your post when I started to post
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  5. #5
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    Quote Originally Posted by skeeter View Post
    do you know how the chain rule works?

    \frac{d}{dx} \left[(\cot^2{x} + x^2)^{\frac{1}{2}}\right]

    \frac{1}{2}(\cot^2{x} + x^2)^{-\frac{1}{2}} \cdot (-2\cot{x}\csc^2{x} + 2x)

    I'll leave the very little bit of possible simplification to you.
    just learned it yesterday.. guess this is good practice. only question. why after you took the derivative of the outermost term, did you put * (-2cotx ...)? i understand that you brought the 2 to the front using the power rule.. but wouldn't it stay positive?

    EDIT: Nevermind, i understand now. Thank you
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  6. #6
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    chain rule inside the chain rule ...

    the derivative of \cot^2{x} is

    2\cot{x} \cdot (-csc^2{x}) = -2\cot^2{x}\csc^2{x}
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  7. #7
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    I think you made a mistake you squared the cot
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