Derivative of 0,2xsin30x-4x

1. Differentiate $\frac{x}{5} \sin{\left(30 x \right)} - 4 x$ term by term:

   1. Apply the quotient rule, which is:

      $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

      $f{\left(x \right)} = x \sin{\left(30 x \right)}$ and $g{\left(x \right)} = 5$.

      To find $\frac{d}{d x} f{\left(x \right)}$:

      1. Apply the product rule:

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

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

         1. Apply the power rule: $x$ goes to $1$

         $g{\left(x \right)} = \sin{\left(30 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

         1. Let $u = 30 x$.

         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 $\frac{d}{d x} 30 x$:

            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: $30$

            The result of the chain rule is:

            $$30 \cos{\left(30 x \right)}$$

         The result is: $30 x \cos{\left(30 x \right)} + \sin{\left(30 x \right)}$

      To find $\frac{d}{d x} g{\left(x \right)}$:

      1. The derivative of the constant $5$ is zero.

      Now plug in to the quotient rule:

      $6 x \cos{\left(30 x \right)} + \frac{\sin{\left(30 x \right)}}{5}$

   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: $-4$

   The result is: $6 x \cos{\left(30 x \right)} + \frac{\sin{\left(30 x \right)}}{5} - 4$

The answer is:

$$6 x \cos{\left(30 x \right)} + \frac{\sin{\left(30 x \right)}}{5} - 4$$