Derivative of -2*sin(x)*cos(x)-1

1. Differentiate $- 2 \sin{\left(x \right)} \cos{\left(x \right)} - 1$ term by term:

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

      1. Apply the product rule:

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

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

         1. The derivative of cosine is negative sine:

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

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

         1. The derivative of sine is cosine:

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

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

      So, the result is: $2 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}$

   2. The derivative of the constant $\left(-1\right) 1$ is zero.

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

2. Now simplify:

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

The answer is:

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