Derivative of ((pi*r)/2)(sin((pi*x)/4))

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

   1. Let $u = \frac{\pi x}{4}$.

   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 $\frac{d}{d x} \frac{\pi x}{4}$:

      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: $\frac{\pi}{4}$

      The result of the chain rule is:

      $$\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{4}$$

   So, the result is: $\frac{\pi^{2} r \cos{\left(\frac{\pi x}{4} \right)}}{8}$

2. Now simplify:

   $$\frac{\pi^{2} r \cos{\left(\frac{\pi x}{4} \right)}}{8}$$

The answer is:

$$\frac{\pi^{2} r \cos{\left(\frac{\pi x}{4} \right)}}{8}$$