Derivative of 4sin(t)*cos(t)

1. Apply the product rule:

   $$\frac{d}{d t} f{\left(t \right)} g{\left(t \right)} = f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}$$

   $f{\left(t \right)} = 4 \sin{\left(t \right)}$; to find $\frac{d}{d t} f{\left(t \right)}$:

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

      1. The derivative of sine is cosine:

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

      So, the result is: $4 \cos{\left(t \right)}$

   $g{\left(t \right)} = \cos{\left(t \right)}$; to find $\frac{d}{d t} g{\left(t \right)}$:

   1. The derivative of cosine is negative sine:

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

   The result is: $- 4 \sin^{2}{\left(t \right)} + 4 \cos^{2}{\left(t \right)}$

2. Now simplify:

   $$4 \cos{\left(2 t \right)}$$

The answer is:

$$4 \cos{\left(2 t \right)}$$