Derivative of (sin(2*x)+1)^2

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

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

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

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

      1. Let $u = 2 x$.

      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 $\frac{d}{d x} 2 x$:

         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: $2$

         The result of the chain rule is:

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

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

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

   The result of the chain rule is:

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

4. Now simplify:

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

The answer is:

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