Derivative of 0.005sin(20pit+p)

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

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

   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 $\frac{\partial}{\partial t} \left(p + 20 \pi t\right)$:

      1. Differentiate $p + 20 \pi t$ term by term:

         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: $20 \pi$

         2. The derivative of the constant $p$ is zero.

         The result is: $20 \pi$

      The result of the chain rule is:

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

   So, the result is: $\frac{\pi \cos{\left(p + 20 \pi t \right)}}{10}$

2. Now simplify:

   $$\frac{\pi \cos{\left(p + 20 \pi t \right)}}{10}$$

The answer is:

$$\frac{\pi \cos{\left(p + 20 \pi t \right)}}{10}$$