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Math Help - Logarithmic Differentiation

  1. #1
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    Logarithmic Differentiation

    4. e) Use Logarithmic differention to find the derivative of the function.

    f(x) = (cos x)^x

    I can turn it into x ln cos x but I do not know how to finish this. Please help.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    f(x) = (cos x)^x

    lnf(x)=ln{(cos x)}^x

    lnf(x)=x*ln(cos x)

    Now we derivative according to x.

    f'(x)/f(x) = ln(cos x) + x*{(-sin x)/(cos x))

    f'(x) = f(x)*{ln(cos x) + x*{(-sin x)/(cos x))}

    f'(x)= {(cos x)^x}*{ln(cos x) + x*{(-sin x)/(cos x))}
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  3. #3
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    y=(\cos x)^x

    Take the ln of both sides.

    ln(y)=ln(\cos x)^x=x \cdot ln(\cos x)

    Differentiate both sides with respect to x.

    \frac{d}{dx}[ln(y)]=\frac{d}{dx}[x \cdot ln(\cos x)]

    Use the chain rule on the left hand side and the product rule on the RHS.

    \frac{1}{y} \cdot \frac{dy}{dx}= ln(\cos x)+x\frac{d}{dx}[ln(\cos x)]

    Now use the chain rule on the RHS.

    \frac{1}{y} \cdot \frac{dy}{dx}=ln(\cos x)+x (\frac{1}{\cos x}) (-\sin x)=ln(\cos x)-x \tan x

    Finally multiply both sides by 'y'.

    \frac{dy}{dx}=y[ln(\cos x)-x \tan x]=(\cos x)^x[ln(\cos x)-x \tan x].
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