Derivative of x*ln(1-2x^2)

1. Apply the product rule:

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

   $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

   $g{\left(x \right)} = \log{\left(1 - 2 x^{2} \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 1 - 2 x^{2}$.

   2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - 2 x^{2}\right)$:

      1. Differentiate $1 - 2 x^{2}$ 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: $x^{2}$ goes to $2 x$

            So, the result is: $- 4 x$

         The result is: $- 4 x$

      The result of the chain rule is:

      $$- \frac{4 x}{1 - 2 x^{2}}$$

   The result is: $- \frac{4 x^{2}}{1 - 2 x^{2}} + \log{\left(1 - 2 x^{2} \right)}$

2. Now simplify:

   $$\frac{4 x^{2} + \left(2 x^{2} - 1\right) \log{\left(1 - 2 x^{2} \right)}}{2 x^{2} - 1}$$

The answer is:

$$\frac{4 x^{2} + \left(2 x^{2} - 1\right) \log{\left(1 - 2 x^{2} \right)}}{2 x^{2} - 1}$$