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Math Help - find f'(pi/4) of f(x) = [ln (cos x)]^2

  1. #1
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    find f'(pi/4) of f(x) = [ln (cos x)]^2

    find f'(pi/4) of f(x) = [ln (cos x)]^2

    I know f'(x) = -2 tan x ln (cos x)

    two questions for f'(pi/4)

    I know the answer is ln 2

    so I know I compute -2 tan pi/4 ln (cos pi/4)

    first question is tan pi/4 * cos pi/4 become 1/sqrt2. is there an easy way of remember such things?

    next the problem then becomes -2 ln (2^-1/2)
    which becomes ln 2. I am missing how it becomes ln 2. ??
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  2. #2
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    Quote Originally Posted by colorado View Post
    find f'(pi/4) of f(x) = [ln (cos x)]^2

    I know f'(x) = -2 tan x ln (cos x)

    two questions for f'(pi/4)

    I know the answer is ln 2

    so I know I compute -2 tan pi/4 ln (cos pi/4)

    first question is tan pi/4 * cos pi/4 become 1/sqrt2. is there an easy way of remember such things?

    next the problem then becomes -2 ln (2^-1/2)
    which becomes ln 2. I am missing how it becomes ln 2. ??
    f(x) = [\ln(\cos{x})]^2

    f'(x) = 2\ln(\cos{x}) \cdot (-\tan{x})

    f'\left(\frac{\pi}{4}\right) = 2\ln\left[\cos\left(\frac{\pi}{4}\right)\right] \cdot \left[-\tan\left(\frac{\pi}{4}\right)\right]

    f'\left(\frac{\pi}{4}\right) = 2\ln\left(\frac{\sqrt{2}}{2}\right) \cdot \left(-1\right) = \ln\left(\frac{\sqrt{2}}{2}\right)^2 \cdot \left(-1\right) = \ln\left(\frac{1}{2}\right) \cdot \left(-1\right) = -\ln(2) \cdot (-1) = \ln(2)
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  3. #3
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    Quote Originally Posted by colorado View Post

    first question is tan pi/4 * cos pi/4 become 1/sqrt2. is there an easy way of remember such things?

    Not this eaxctly, but you do have to remember that \displaystyle\tan \frac{\pi}{4} = 1 and \displaystyle\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}

    Quote Originally Posted by colorado View Post


    next the problem then becomes -2 ln (2^-1/2)
    which becomes ln 2. I am missing how it becomes ln 2. ??
    \displaystyle -2 \ln 2^{\frac{-1}{2}} = \frac{-1}{2}\times -2 \ln 2 = \ln 2
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  4. #4
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    is this from the log property loga (x^y) = y loga x

    meaning ln 2^-1/2 = -1/2 ln 2?
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