Derivative of cos^23x

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

2. Apply the power rule: $u^{23}$ goes to $23 u^{22}$

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

   1. The derivative of cosine is negative sine:

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

   The result of the chain rule is:

   $$- 23 \sin{\left(x \right)} \cos^{22}{\left(x \right)}$$

The answer is:

$$- 23 \sin{\left(x \right)} \cos^{22}{\left(x \right)}$$