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Thread: Inverse Cosine Derivative

  1. #1
    Member RedBarchetta's Avatar
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    Inverse Cosine Derivative

    This is stumping me. Find the derivative of the following function:

    $\displaystyle
    y = \cos ^{ - 1} \left( {\frac{x}
    {2}} \right)
    $

    We know this rule:

    $\displaystyle
    \frac{d}
    {{dx}}\cos ^{ - 1} u = \frac{{ - du/dx}}
    {{\sqrt {1 - u^2 } }},|u| < 1
    $

    So.....


    $\displaystyle
    \begin{gathered}
    u = \tfrac{x}
    {2} \hfill \\
    \tfrac{{du}}
    {{dx}} = \tfrac{1}
    {2} \hfill \\
    \frac{d}
    {{dx}}\cos ^{ - 1} u = \frac{{ - 1/2}}
    {{\sqrt {1 - \tfrac{{x^2 }}
    {4}} }} = \frac{{ - 1/2}}
    {{\sqrt {4 - x^2 } }} \hfill \\
    \end{gathered}
    $

    But this is wrong....it should be....

    $\displaystyle
    y = \cos ^{ - 1} \left( {\frac{x}
    {2}} \right) = \frac{{ - 1}}
    {{\sqrt {4 - x^2 } }}
    $

    What's going on here?
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  2. #2
    Super Member 11rdc11's Avatar
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    $\displaystyle \frac{{ - 1/2}}
    {{\sqrt {1 - \tfrac{{x^2 }}
    {4}} }}$

    Multipy that by

    $\displaystyle \frac{\sqrt{4}}{\sqrt{4}}$
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  3. #3
    Member RedBarchetta's Avatar
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    Quote Originally Posted by 11rdc11 View Post
    $\displaystyle \frac{{ - 1/2}}
    {{\sqrt {1 - \tfrac{{x^2 }}
    {4}} }}$

    Multipy that by

    $\displaystyle \frac{\sqrt{4}}{\sqrt{4}}$
    So this is true as well? Given a as a constant.

    $\displaystyle
    \frac{d}
    {{dx}}\cos ^{ - 1} \frac{x}
    {a} = \frac{{ - 1}}
    {{\sqrt {a^2 - x^2 } }}
    $
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  4. #4
    Super Member 11rdc11's Avatar
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    I think so
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  5. #5
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by RedBarchetta View Post
    So this is true as well? Given a as a constant.

    $\displaystyle
    \frac{d}
    {{dx}}\cos ^{ - 1} \frac{x}
    {a} = \frac{{ - 1}}
    {{\sqrt {a^2 - x^2 } }}
    $

    Yup!

    Because $\displaystyle \frac{d}{\,dx}\cos^{-1}\left(\frac{x}{a}\right)=-\frac{1}{a}\frac{1}{\sqrt{1-\frac{x^2}{a^2}}}=-\frac{1}{\sqrt{a^2}}\frac{1}{\sqrt{1-\frac{x^2}{a^2}}}=-\frac{1}{\sqrt{a^2\left(1-\frac{x^2}{a^2}\right)}}=-\frac{1}{\sqrt{a^2-x^2}}$

    --Chris
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