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Math Help - logarithmic differentiation

  1. #1
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    Question logarithmic differentiation

    Find the derivative of

    y=(1+cosx)^(2x)

    by taking the ln of both sides then using implicit differentiation.

    can someone show me the steps for this?
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  2. #2
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    Hello, thecount!

    Find the derivative of: . y \:=\:(1+\cos x)^{2x}

    by taking the \ln of both sides then using implicit differentiation.

    Take the \ln of both sides: . \ln(y) \:=\:\ln(1 + \cos x)^{2x} \quad\Rightarrow\quad \ln(y) \:=\:2x\cdot\ln(1 + \cos x)


    Differentiate implicitly (product rule):

    . . \frac{1}{y}\,\frac{dy}{dx} \;=\;2x\cdot\frac{1}{1+\cos x}\cdot(-\sin x) + 2\cdot\ln(1+\cos x) \;=\;\frac{-2\sin x}{1+\cos x} + 2\ln(1+\cos x)

    Multiply by y\!:\;\;\frac{dy}{dx} \;=\;y\cdot\bigg[\frac{-2\sin x}{1 + \cos x} + 2\ln(1+\cos x)\bigg]


    Therefore: . \frac{dy}{dx} \;=\;(1+\cos x)^{2x}\left[\frac{-2\sin x}{1 + \cos x} + 2\ln(1 + \cos x)\right]

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