Derivative of 2x+5(cos(2x)^3)

1. Differentiate $2 x + 5 \cos^{3}{\left(2 x \right)}$ 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 a constant times a function is the constant times the derivative of the function.

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

      2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

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

         1. Let $u = 2 x$.

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

            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:

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

         The result of the chain rule is:

         $$- 6 \sin{\left(2 x \right)} \cos^{2}{\left(2 x \right)}$$

      So, the result is: $- 30 \sin{\left(2 x \right)} \cos^{2}{\left(2 x \right)}$

   The result is: $- 30 \sin{\left(2 x \right)} \cos^{2}{\left(2 x \right)} + 2$

The answer is:

$$- 30 \sin{\left(2 x \right)} \cos^{2}{\left(2 x \right)} + 2$$