Derivative of sin^9(x/2)

1. Let $u = \sin{\left(\frac{x}{2} \right)}$.

2. Apply the power rule: $u^{9}$ goes to $9 u^{8}$

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

   1. Let $u = \frac{x}{2}$.

   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 $\frac{d}{d x} \frac{x}{2}$:

      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: $\frac{1}{2}$

      The result of the chain rule is:

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

   The result of the chain rule is:

   $$\frac{9 \sin^{8}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{2}$$

4. Now simplify:

   $$\frac{9 \sin^{8}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{2}$$

The answer is: