Derivative of -5*sin^2(pi*x/6)

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

   1. Let $u = \sin{\left(\frac{\pi x}{6} \right)}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\frac{\pi x}{6} \right)}$:

      1. Let $u = \frac{\pi x}{6}$.

      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 $\frac{d}{d x} \frac{\pi x}{6}$:

         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: $x$ goes to $1$

               So, the result is: $\pi$

            So, the result is: $\frac{\pi}{6}$

         The result of the chain rule is:

         $$\frac{\pi \cos{\left(\frac{\pi x}{6} \right)}}{6}$$

      The result of the chain rule is:

      $$\frac{\pi \sin{\left(\frac{\pi x}{6} \right)} \cos{\left(\frac{\pi x}{6} \right)}}{3}$$

   So, the result is: $- \frac{5 \pi \sin{\left(\frac{\pi x}{6} \right)} \cos{\left(\frac{\pi x}{6} \right)}}{3}$

2. Now simplify:

   $$- \frac{5 \pi \sin{\left(\frac{\pi x}{3} \right)}}{6}$$

The answer is:

$$- \frac{5 \pi \sin{\left(\frac{\pi x}{3} \right)}}{6}$$