Derivative of 5/3*sin(3*x)-5*x

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

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

      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:

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

      So, the result is: $5 \cos{\left(3 x \right)}$

   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: $-5$

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

The answer is:

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