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Find the derivative of (x2(3-x)1/3)/((1-x)(3+x)2/3)

Find the derivative of the function

    \[ f(x) = \frac{x^2(3-x)^{\frac{1}{3}}}{(1-x)(3+x)^{\frac{2}{3}}}. \]


First, we take the logarithm of both sides so we can use logarithmic differentiation,

    \[ \log f(x) = 2 \log x + \frac{1}{3} \log (3-x) - \log (1-x) - \frac{2}{3} \log (3+x). \]

Then we differentiate both sides,

    \begin{align*}  \frac{f'(x)}{f(x)} &= \frac{2}{x} + \frac{1}{3(3-x)} + \frac{1}{1-x} - \frac{2}{3(3+x)} \\[9pt]  &= \frac{18-7x}{3x(3-x)} + \frac{7+5x}{3(1-x)(3+x)} \\[9pt]  &= \frac{(54 - 57x - 4x^2 + 7x^3) + (21x + 8x^2 - 5x^3)}{3x(3-x)(1-x)(3+x)} \\[9pt]  &= \frac{2x^3 + 4x^2 - 36x + 54}{3x(1-x)(3-x)(3+x)}. \end{align*}

Therefore,

    \begin{align*}  f'(x) &= \left( \frac{x^2(3-x)^{\frac{1}{3}}}{(1-x)(3+x)^{\frac{2}{3}}} \right) \left( \frac{2x^3 + 4x^2 - 36x + 54}{3x(1-x)(3-x)(3+x)} \right)\\[9pt]  &= \frac{2x^4 + 4x^3 - 36x^2 + 54x}{3(1-x)^2(3-x)^{\frac{2}{3}} (3+x)^{\frac{5}{3}}}. \end{align*}

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