Derivative of sin(x)^2*(cos(x)+1)

1. Apply the product rule:

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

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

   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:

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

   $g{\left(x \right)} = \cos{\left(x \right)} + 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $\cos{\left(x \right)} + 1$ term by term:

      1. The derivative of cosine is negative sine:

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

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

      The result is: $- \sin{\left(x \right)}$

   The result is: $2 \left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} - \sin^{3}{\left(x \right)}$

2. Now simplify:

   $$\left(- 3 \sin^{2}{\left(x \right)} + 2 \cos{\left(x \right)} + 2\right) \sin{\left(x \right)}$$

The answer is:

$$\left(- 3 \sin^{2}{\left(x \right)} + 2 \cos{\left(x \right)} + 2\right) \sin{\left(x \right)}$$