Derivative of 4sin2t

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

   1. Let $u = 2 t$.

   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 $\frac{d}{d t} 2 t$:

      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: $2$

      The result of the chain rule is:

      $$2 \cos{\left(2 t \right)}$$

   So, the result is: $8 \cos{\left(2 t \right)}$

The answer is:

$$8 \cos{\left(2 t \right)}$$