Derivative of 8*pi*sin(pi*x)

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

   1. Let $u = \pi x$.

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

      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: $\pi$

      The result of the chain rule is:

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

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

The answer is:

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