Derivative of sin(2pi*z)

1. Let $u = 2 \pi z$.

2. The derivative of sine is cosine:

   $$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$$

3. Then, apply the chain rule. Multiply by $\frac{d}{d z} 2 \pi z$:

   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: $2 \pi$

   The result of the chain rule is:

   $$2 \pi \cos{\left(2 \pi z \right)}$$

4. Now simplify:

   $$2 \pi \cos{\left(2 \pi z \right)}$$

The answer is:

$$2 \pi \cos{\left(2 \pi z \right)}$$