Derivative of \( \displaystyle\frac{x^{3}}{1-x^{3}} \) w.r.t. \( x^{3} \), \( x\ne 0,1 \) is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{1}{(1-x^{3})^{2}} \)
Let \( t=x^{3} \). The function becomes \( \frac{t}{1-t} \), and we need its derivative with respect to t.
By the quotient rule,
\( \displaystyle \frac{d}{dt}\left(\frac{t}{1-t}\right)=\frac{(1-t)+t}{(1-t)^{2}} \)
\( \displaystyle =\frac{1}{(1-t)^{2}} \)
Putting back \( t=x^{3} \):
\( \displaystyle \frac{1}{(1-x^{3})^{2}} \)