Derivative of sin(2x)+cos(4x)

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

   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:

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

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

The answer is:

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