Derivative of cos(4pi*t)

1. Let $u = 4 \pi t$.

2. The derivative of cosine is negative sine:

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d t} 4 \pi t$:

   1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: $t$ goes to $1$

      So, the result is: $4 \pi$

   The result of the chain rule is:

   $$- 4 \pi \sin{\left(4 \pi t \right)}$$

The answer is: