Derivative of (sin^8x)

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

2. Apply the power rule: $u^{8}$ goes to $8 u^{7}$

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:

   $$8 \sin^{7}{\left(x \right)} \cos{\left(x \right)}$$

The answer is:

$$8 \sin^{7}{\left(x \right)} \cos{\left(x \right)}$$