Derivative of x^2/(1-x)

1. Apply the quotient rule, which is:

   $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

   $f{\left(x \right)} = x^{2}$ and $g{\left(x \right)} = 1 - x$.

   To find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x^{2}$ goes to $2 x$

   To find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $1 - x$ term by term:

      1. The derivative of the constant $1$ is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $-1$

      The result is: $-1$

   Now plug in to the quotient rule:

   $\frac{x^{2} + 2 x \left(1 - x\right)}{\left(1 - x\right)^{2}}$

2. Now simplify:

   $$\frac{x \left(2 - x\right)}{\left(x - 1\right)^{2}}$$

The answer is:

$$\frac{x \left(2 - x\right)}{\left(x - 1\right)^{2}}$$