Derivative of 2xsin4x+cos4x

1. Differentiate $2 x \sin{\left(4 x \right)} + \cos{\left(4 x \right)}$ term by term:

   1. Apply the product rule:

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

      $f{\left(x \right)} = 2 x$; 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. Apply the power rule: $x$ goes to $1$

         So, the result is: $2$

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

      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:

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

      The result is: $8 x \cos{\left(4 x \right)} + 2 \sin{\left(4 x \right)}$

   2. Let $u = 4 x$.

   3. The derivative of cosine is negative sine:

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

   4. 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: $8 x \cos{\left(4 x \right)} - 2 \sin{\left(4 x \right)}$

The answer is:

$$8 x \cos{\left(4 x \right)} - 2 \sin{\left(4 x \right)}$$