Derivative of cbrt(sqrt(cos(3x+2)))

1. Let $u = \sqrt{\cos{\left(3 x + 2 \right)}}$.

2. Apply the power rule: $\sqrt[3]{u}$ goes to $\frac{1}{3 u^{\frac{2}{3}}}$

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

   1. Let $u = \cos{\left(3 x + 2 \right)}$.

   2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$

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

      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:

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

      The result of the chain rule is:

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

   The result of the chain rule is:

   $$- \frac{\sin{\left(3 x + 2 \right)}}{2 \cos^{\frac{5}{6}}{\left(3 x + 2 \right)}}$$

4. Now simplify:

   $$- \frac{\sin{\left(3 x + 2 \right)}}{2 \cos^{\frac{5}{6}}{\left(3 x + 2 \right)}}$$

The answer is:

$$- \frac{\sin{\left(3 x + 2 \right)}}{2 \cos^{\frac{5}{6}}{\left(3 x + 2 \right)}}$$