Derivative of sin^2z

1. Let $u = \sin{\left(z \right)}$.

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

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

   1. The derivative of sine is cosine:

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

   The result of the chain rule is:

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

4. Now simplify:

   $$\sin{\left(2 z \right)}$$

The answer is:

$$\sin{\left(2 z \right)}$$