Derivative of -sin(x)*(1+cos(x))

1. Apply the product rule:

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

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

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

      1. The derivative of sine is cosine:

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

      So, the result is: $- \cos{\left(x \right)}$

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

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

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

      2. The derivative of cosine is negative sine:

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

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

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

2. Now simplify:

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

The answer is:

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