Derivative of f(x)=sin(3x)+5x+2

1. Differentiate $\left(5 x + \sin{\left(3 x \right)}\right) + 2$ term by term:

   1. Differentiate $5 x + \sin{\left(3 x \right)}$ term by term:

      1. Let $u = 3 x$.

      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 $\frac{d}{d x} 3 x$:

         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: $3$

         The result of the chain rule is:

         $$3 \cos{\left(3 x \right)}$$

      4. 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: $5$

      The result is: $3 \cos{\left(3 x \right)} + 5$

   2. The derivative of the constant $2$ is zero.

   The result is: $3 \cos{\left(3 x \right)} + 5$

The answer is:

$$3 \cos{\left(3 x \right)} + 5$$