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Math Help - Derivative

  1. #1
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    Derivative

    Evaluate the derivative of the function below at the point (, ).

    y=7/x + sqrt cos(x)
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  2. #2
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    well the derivative of that function is just
    -7/(x^2)+[1/(2(sqrt(cos(x)))]*-sin(x)

    evaluation of derivative of sqrt[cos(x)] is just applying the chain rule.
    now all u have to do is substitute the value of x in.
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  3. #3
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    This is the part I don't get...actually putting TT/4 into the equation...

    I know cos(x) = sqrt2/2 and the same for sin (x), right?
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  4. #4
    Member sinewave85's Avatar
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    Quote Originally Posted by tradar View Post
    Evaluate the derivative of the function below at the point (, ).

    y=7/x + sqrt cos(x)
    First, rewrite the function like this:
    y = 7x^{-1} + (\cos{x})^{\frac{1}{2}}

    then, by the chain rule:

    y^{\prime} = -7x^{-2} + \frac{1}{2}(cos{x})^{-\frac{1}{2}}(-\sin{x})

    and y^{\prime}\left(\frac{\pi}{4}\right) = -11.76844:
    • \frac{\pi}{4} is not a critical number
    • the curve is falling at \left(\frac{\pi}{4}, 9.754\right)
    Last edited by sinewave85; March 21st 2009 at 06:54 AM.
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  5. #5
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    Quote Originally Posted by tradar View Post
    Evaluate the derivative of the function below at the point (, ).

    y=7/x + sqrt cos(x)
     \frac{dy}{dx}=\frac{-7}{x^2}-\frac{1}{2}(cosx)^{-\frac{1}{2}}.sinx

     \frac{dy}{dx}=\frac{-7}{(\frac{\pi}{4})^2}-\frac{1}{2}(cos(\frac{\pi}{4}))^{-\frac{1}{2}}.sin(\frac{\pi}{4}) =-11.77
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  6. #6
    Member sinewave85's Avatar
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    Quote Originally Posted by tradar View Post
    This is the part I don't get...actually putting TT/4 into the equation...

    I know cos(x) = sqrt2/2 and the same for sin (x), right?
    Right. Just to clarify:


    <br />
y^{\prime} = -7x^{-2} + \frac{1}{2}(cos{\frac{\pi}{4}})^{-\frac{1}{2}}(-\sin{\frac{\pi}{4}}) = -7\left(\frac{\pi}{4}\right)^{-2} + \frac{1}{2}\left(\frac{\sqrt{2}}{2}\right)^{-\frac{1}{2}}\left(-\frac{\sqrt{2}}{2}\right) = -11.7684<br />
    Last edited by sinewave85; March 21st 2009 at 06:55 AM.
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  7. #7
    Member u2_wa's Avatar
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    Quote Originally Posted by sinewave85 View Post
    Right. Just to clarify:


    <br />
y^{\prime} = -7\color{red}{x}\color{black}^{-2} + \frac{1}{2}(cos{\frac{\pi}{4}})^{-\frac{1}{2}}(-\sin{\frac{\pi}{4}}) = -7\color{red}{x}\color{black}^{-2} + \frac{1}{2}\left(\frac{\sqrt{2}}{2}\right)^{-\frac{1}{2}}\left(-\frac{\sqrt{2}}{2}\right) = -0.4408563709<br />
    I think there is a mistake!
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  8. #8
    Member sinewave85's Avatar
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    Quote Originally Posted by u2_wa View Post
    I think there is a mistake!
    Big oops. Thanks. I somehow lost that x as I went along.
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