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Compute the derivative of the given function

Compute the derivative of the function

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

where x^3 \neq 3.


We compute this directly, (we can use this exercise to quickly get the formula for the derivative of a product of three differentiable functions, although it’s not really necessary since it is still a direct application of the product rule for derivatives)

    \begin{align*}  f'(x) &= (1+x)' (2+x^2)^{\frac{1}{2}} (3+x^3)^{\frac{1}{3}} + (1+x)\left( (2+x^2)^{\frac{1}{2}} \right)' (3 + x^3)^{\frac{1}{3}} \\ & \qquad \qquad + (1+x)(2+x^2)^{\frac{1}{2}}\left( (3+x^3)^{\frac{1}{3}} \right)' \\  &= (2+x^2)^{\frac{1}{2}} (3+x^3)^{\frac{1}{3}} + (1+x)(3+x^3)^{\frac{1}{3}} \frac{x}{\sqrt{2+x^2}} + (1+x)(2+x^2)^{\frac{1}{2}} \frac{x^2}{(3+x^3)^{\frac{2}{3}}} \\[9pt]  &= \frac{3x^5 + 2x^4 + 4x^3 + 8x^2 + 3x + 6}{(2+x^2)^{\frac{1}{2}} (3 + x^3)^{\frac{2}{3}}} \end{align*}

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