Derivative of 5*cos(4x+2)+9

1. Differentiate $5 \cos{\left(4 x + 2 \right)} + 9$ term by term:

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

      1. Let $u = 4 x + 2$.

      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 $\frac{d}{d x} \left(4 x + 2\right)$:

         1. Differentiate $4 x + 2$ 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: $4$

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

            The result is: $4$

         The result of the chain rule is:

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

      So, the result is: $- 20 \sin{\left(4 x + 2 \right)}$

   2. The derivative of the constant $9$ is zero.

   The result is: $- 20 \sin{\left(4 x + 2 \right)}$

2. Now simplify:

   $$- 20 \sin{\left(4 x + 2 \right)}$$

The answer is:

$$- 20 \sin{\left(4 x + 2 \right)}$$