Derivative of x(sin(4x)+cos(4x))

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 $1$

   $g{\left(x \right)} = \sin{\left(4 x \right)} + \cos{\left(4 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $\sin{\left(4 x \right)} + \cos{\left(4 x \right)}$ term by term:

      1. Let $u = 4 x$.

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

         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: $4$

         The result of the chain rule is:

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

      4. Let $u = 4 x$.

      5. The derivative of cosine is negative sine:

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

      6. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$:

         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: $4$

         The result of the chain rule is:

         $$- 4 \sin{\left(4 x \right)}$$

      The result is: $- 4 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}$

   The result is: $x \left(- 4 \sin{\left(4 x \right)} + 4 \cos{\left(4 x \right)}\right) + \sin{\left(4 x \right)} + \cos{\left(4 x \right)}$

2. Now simplify:

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

The answer is:

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