Derivative of cos^3t

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

2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

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

   1. The derivative of cosine is negative sine:

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

   The result of the chain rule is:

   $$- 3 \sin{\left(t \right)} \cos^{2}{\left(t \right)}$$

The answer is:

$$- 3 \sin{\left(t \right)} \cos^{2}{\left(t \right)}$$