Derivative of 4cos(3x+2)

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

   1. Let $u = 3 x + 2$.

   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 $\frac{d}{d x} \left(3 x + 2\right)$:

      1. Differentiate $3 x + 2$ 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: $3$

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

         The result is: $3$

      The result of the chain rule is:

      $$- 3 \sin{\left(3 x + 2 \right)}$$

   So, the result is: $- 12 \sin{\left(3 x + 2 \right)}$

2. Now simplify:

   $$- 12 \sin{\left(3 x + 2 \right)}$$

The answer is:

$$- 12 \sin{\left(3 x + 2 \right)}$$