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

1. Apply the product rule:

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

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

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

      The result is: $- 2 x$

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

   1. Let $u = \sin{\left(x \right)}$.

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

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

      1. The derivative of sine is cosine:

         $$\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$$

      The result of the chain rule is:

      $$2 \sin{\left(x \right)} \cos{\left(x \right)}$$

   The result is: $- 2 x \sin^{2}{\left(x \right)} + 2 \cdot \left(1 - x^{2}\right) \sin{\left(x \right)} \cos{\left(x \right)}$

2. Now simplify:

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

The answer is:

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