Derivative of cos^7x

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

2. Apply the power rule: $u^{7}$ goes to $7 u^{6}$

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:

   $$- 7 \sin{\left(x \right)} \cos^{6}{\left(x \right)}$$

The answer is:

$$- 7 \sin{\left(x \right)} \cos^{6}{\left(x \right)}$$