Derivative of cosx+sin2x+3x

1. Differentiate $3 x + \left(\sin{\left(2 x \right)} + \cos{\left(x \right)}\right)$ term by term:

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

      1. The derivative of cosine is negative sine:

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

      2. Let $u = 2 x$.

      3. The derivative of sine is cosine:

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

      4. 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:

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

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

   2. 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: $3$

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

The answer is:

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