Derivative of sin(2*x-12*pi/11)+2*cos(x-6*pi/11)

1. Differentiate $\sin{\left(2 x - \frac{12 \pi}{11} \right)} + 2 \cos{\left(x - \frac{6 \pi}{11} \right)}$ term by term:

   1. Let $u = 2 x - \frac{12 \pi}{11}$.

   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 $\frac{d}{d x} \left(2 x - \frac{12 \pi}{11}\right)$:

      1. Differentiate $2 x - \frac{12 \pi}{11}$ 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 $- \frac{12 \pi}{11}$ is zero.

         The result is: $2$

      The result of the chain rule is:

      $$2 \cos{\left(2 x - \frac{12 \pi}{11} \right)}$$

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

      1. Let $u = x - \frac{6 \pi}{11}$.

      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 $\frac{d}{d x} \left(x - \frac{6 \pi}{11}\right)$:

         1. Differentiate $x - \frac{6 \pi}{11}$ term by term:

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

            2. The derivative of the constant $- \frac{6 \pi}{11}$ is zero.

            The result is: $1$

         The result of the chain rule is:

         $$- \sin{\left(x - \frac{6 \pi}{11} \right)}$$

      So, the result is: $- 2 \sin{\left(x - \frac{6 \pi}{11} \right)}$

   The result is: $- 2 \sin{\left(x - \frac{6 \pi}{11} \right)} + 2 \cos{\left(2 x - \frac{12 \pi}{11} \right)}$

2. Now simplify:

   $$2 \sin{\left(x + \frac{5 \pi}{11} \right)} - 2 \sin{\left(2 x + \frac{9 \pi}{22} \right)}$$

The answer is:

$$2 \sin{\left(x + \frac{5 \pi}{11} \right)} - 2 \sin{\left(2 x + \frac{9 \pi}{22} \right)}$$