Derivative of sin^2(cos3x)+2pi

1. Differentiate $\sin^{2}{\left(\cos{\left(3 x \right)} \right)} + 2 \pi$ term by term:

   1. Let $u = \sin{\left(\cos{\left(3 x \right)} \right)}$.

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

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

      1. Let $u = \cos{\left(3 x \right)}$.

      2. The derivative of sine is cosine:

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

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

         1. Let $u = 3 x$.

         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 $\frac{d}{d x} 3 x$:

            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: $3$

            The result of the chain rule is:

            $$- 3 \sin{\left(3 x \right)}$$

         The result of the chain rule is:

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

      The result of the chain rule is:

      $$- 6 \sin{\left(3 x \right)} \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(\cos{\left(3 x \right)} \right)}$$

   4. The derivative of the constant $2 \pi$ is zero.

   The result is: $- 6 \sin{\left(3 x \right)} \sin{\left(\cos{\left(3 x \right)} \right)} \cos{\left(\cos{\left(3 x \right)} \right)}$

2. Now simplify:

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

The answer is: