Derivative of x(a*cos2x+bsin2x)

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 $1$

   $g{\left(x \right)} = a \cos{\left(2 x \right)} + b \sin{\left(2 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $a \cos{\left(2 x \right)} + b \sin{\left(2 x \right)}$ term by term:

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

         1. Let $u = 2 x$.

         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 $\frac{d}{d x} 2 x$:

            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: $2$

            The result of the chain rule is:

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

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

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

         1. Let $u = 2 x$.

         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 $\frac{d}{d x} 2 x$:

            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: $2$

            The result of the chain rule is:

            $$2 \cos{\left(2 x \right)}$$

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

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

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

2. Now simplify:

   $$a \cos{\left(2 x \right)} + b \sin{\left(2 x \right)} - 2 x \left(a \sin{\left(2 x \right)} - b \cos{\left(2 x \right)}\right)$$

The answer is:

$$a \cos{\left(2 x \right)} + b \sin{\left(2 x \right)} - 2 x \left(a \sin{\left(2 x \right)} - b \cos{\left(2 x \right)}\right)$$