Derivative of y=sin^9x

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

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

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

   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:

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

The answer is:

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