Derivative of 2sin(2x)+(3+3cos4x)

1. Differentiate $2 \sin{\left(2 x \right)} + 3 \cos{\left(4 x \right)} + 3$ term by term:

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

      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:

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

      So, the result is: $4 \cos{\left(2 x \right)}$

   2. The derivative of the constant $3$ is zero.

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

      1. Let $u = 4 x$.

      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 $\frac{d}{d x} 4 x$:

         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: $4$

         The result of the chain rule is:

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

      So, the result is: $- 12 \sin{\left(4 x \right)}$

   The result is: $- 12 \sin{\left(4 x \right)} + 4 \cos{\left(2 x \right)}$

The answer is:

$$- 12 \sin{\left(4 x \right)} + 4 \cos{\left(2 x \right)}$$