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

1. Differentiate $1 - 2 \cos^{2}{\left(x \right)}$ 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. Let $u = \cos{\left(x \right)}$.

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

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

         1. The derivative of cosine is negative sine:

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

         The result of the chain rule is:

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

      So, the result is: $4 \sin{\left(x \right)} \cos{\left(x \right)}$

   The result is: $4 \sin{\left(x \right)} \cos{\left(x \right)}$

2. Now simplify:

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

The answer is:

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