Derivative of cos(2*z+3*i)^(2)

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

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

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

   1. Let $u = 2 z + 3 i$.

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

      1. Differentiate $2 z + 3 i$ 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: $z$ goes to $1$

            So, the result is: $2$

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

         The result is: $2$

      The result of the chain rule is:

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

   The result of the chain rule is:

   $$- 4 \sin{\left(2 z + 3 i \right)} \cos{\left(2 z + 3 i \right)}$$

The answer is:

$$- 4 \sin{\left(2 z + 3 i \right)} \cos{\left(2 z + 3 i \right)}$$