Derivative of 5*sin(20pit)

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

   1. Let $u = 20 \pi t$.

   2. The derivative of sine is cosine:

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d t} 20 \pi t$:

      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: $20 \pi$

      The result of the chain rule is:

      $$20 \pi \cos{\left(20 \pi t \right)}$$

   So, the result is: $100 \pi \cos{\left(20 \pi t \right)}$

The answer is:

$$100 \pi \cos{\left(20 \pi t \right)}$$