Derivative of 4x(cosx^3)

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

   1. Apply the product rule:

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

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

      1. Apply the power rule: $x$ goes to $1$

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

      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:

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

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

   So, the result is: $- 12 x \sin{\left(x \right)} \cos^{2}{\left(x \right)} + 4 \cos^{3}{\left(x \right)}$

2. Now simplify:

   $$4 \left(- 3 x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}$$

The answer is:

$$4 \left(- 3 x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}$$