Differentiation of (x^3)/(1-x^3) with respect to x^3 is equal to:

The correct answer is Option (1) → \(\displaystyle\frac{1}{(1-x^3)^2},x \ne 1\)

Let \(u = x^3\). Then the function is \(\displaystyle\frac{u}{1-u}\), to be differentiated with respect to u. By the quotient rule:

    \( \displaystyle \frac{d}{du}\left(\frac{u}{1-u}\right) = \frac{(1-u)(1) - u(-1)}{(1-u)^2} \)

    \( \displaystyle = \frac{1}{(1-u)^2} \)

    \( \displaystyle = \frac{1}{(1-x^3)^2} \)

The expression is defined only when \(x^3 \ne 1\), that is \(x \ne 1\).