Home » Blog » Compute the derivative of the given function

Compute the derivative of the given function

Compute the derivative of the function

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


Using the chain rule we compute,

    \begin{align*}  f'(x) &= \frac{1}{3} \left( \frac{1+x^3}{1-x^3} \right)^{-\frac{2}{3}} \left( \frac{3x^2(1-x^3) - (1+x^3)(-3x^2)}{(1-x^3)^2} \right) \\  &= \frac{1}{3} \left( \frac{1+x^3}{1-x^3} \right)^{-\frac{2}{3}} \left( \frac{3x^2 - 3x^5 + 3x^2 + 3x^5}{(1-x^3)^2} \right) \\  &= \frac{1}{3} \left( \frac{1+x^3}{1-x^3} \right)^{-\frac{2}{3}} \left( \frac{6x^2}{(1-x^3)^2} \right) \\  &= \frac{2x^2}{1-x^3} \frac{1}{1-x^3} \left( \frac{1+x^3}{1-x^3} \right)^{-\frac{2}{3}} \\  &= \frac{2x^2 (1+x^3)}{(1-x^3)(1+x^3)} \frac{1}{1-x^3} \left( \frac{1+x^3}{1-x^3} \right)^{-\frac{2}{3}} \\  &= \frac{2x^2}{1-x^6} \frac{1+x^3}{1-x^3} \left( \frac{1 + x^3}{1-x^3} \right)^{-\frac{2}{3}} \\  &= \frac{2x^2}{1-x^6} \left( \frac{1+x^3}{1-x^3} \right)^{\frac{1}{3}}. \end{align*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):