Derivative of 4*sin^2((pi*t)/3)

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

   1. Let $u = \sin{\left(\frac{\pi t}{3} \right)}$.

   2. Apply the power rule: $u^{2}$ goes to $2 u$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d t} \sin{\left(\frac{\pi t}{3} \right)}$:

      1. Let $u = \frac{\pi t}{3}$.

      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 $\frac{d}{d t} \frac{\pi t}{3}$:

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

            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: $\pi$

            So, the result is: $\frac{\pi}{3}$

         The result of the chain rule is:

         $$\frac{\pi \cos{\left(\frac{\pi t}{3} \right)}}{3}$$

      The result of the chain rule is:

      $$\frac{2 \pi \sin{\left(\frac{\pi t}{3} \right)} \cos{\left(\frac{\pi t}{3} \right)}}{3}$$

   So, the result is: $\frac{8 \pi \sin{\left(\frac{\pi t}{3} \right)} \cos{\left(\frac{\pi t}{3} \right)}}{3}$

2. Now simplify:

   $$\frac{4 \pi \sin{\left(\frac{2 \pi t}{3} \right)}}{3}$$

The answer is:

$$\frac{4 \pi \sin{\left(\frac{2 \pi t}{3} \right)}}{3}$$