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Math Help - Logarithmic Differentiation

  1. #1
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    Logarithmic Differentiation

    Find f'(x) using logarithmic differentiation:
    f(x)=(4x+4)^(3x)
    Thanks for the help.
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  2. #2
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    Hello, nystudent2729!

    Find f'(x) using logarithmic differentiation: . f(x)\:=\:(4x+4)^{3x}
    We have: . y \;=\;(4x+4)^{3x}

    Take logs: . \ln(y) \;=\;\ln(4x+4)^{3x}

    . . . . . . . . \ln(y) \;=\;3x\!\cdot\!\ln(4x+4)


    Differentiate: . \frac{1}{y}\!\cdot\!\frac{dy}{dx} \;=\;3x\!\cdot\!\frac{4}{4x+4} + 3\!\cdot\!\ln(4x+4) \;=\;3\bigg[\frac{x}{x+1} + \ln(4x+4)\bigg]

    . . Then: . \frac{dy}{dx} \;=\;3y\bigg[\frac{x}{x+1} + \ln(4x+4)\bigg]


    Therefore: . \frac{dy}{dx} \;=\;3(4x+4)^{3x}\left[\frac{x}{x+1} + \ln(4x+4)\right]

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