Derivative of cos^2(2x+1)

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

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(2 x + 1 \right)}$:

   1. Let $u = 2 x + 1$.

   2. The derivative of cosine is negative sine:

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 1\right)$:

      1. Differentiate $2 x + 1$ term by term:

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

            1. Apply the power rule: $x$ goes to $1$

            So, the result is: $2$

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

         The result is: $2$

      The result of the chain rule is:

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

   The result of the chain rule is:

   $$- 4 \sin{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}$$

4. Now simplify:

   $$- 2 \sin{\left(4 x + 2 \right)}$$

The answer is:

$$- 2 \sin{\left(4 x + 2 \right)}$$