Derivative of y=1\2sin²x

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

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

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

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

      1. The derivative of sine is cosine:

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

      The result of the chain rule is:

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

   So, the result is: $\sin{\left(x \right)} \cos{\left(x \right)}$

2. Now simplify:

   $$\frac{\sin{\left(2 x \right)}}{2}$$

The answer is:

$$\frac{\sin{\left(2 x \right)}}{2}$$