Derivative of 9cos^3xcospix

1. Apply the product rule:

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

   $f{\left(x \right)} = 9 \cos^{3}{\left(x \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

      1. Let $u = \cos{\left(x \right)}$.

      2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

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

         1. The derivative of cosine is negative sine:

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

         The result of the chain rule is:

         $$- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$$

      So, the result is: $- 27 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$

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

   1. Let $u = \pi x$.

   2. The derivative of cosine is negative sine:

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

   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:

      $$- \pi \sin{\left(\pi x \right)}$$

   The result is: $- 27 \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\pi x \right)} - 9 \pi \sin{\left(\pi x \right)} \cos^{3}{\left(x \right)}$

2. Now simplify:

   $$- 9 \left(3 \sin{\left(x \right)} \cos{\left(\pi x \right)} + \pi \sin{\left(\pi x \right)} \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}$$

The answer is: