Derivative of А*(x/3)*sin(x/3)

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = a x \sin{\left(\frac{x}{3} \right)}$ and $g{\left(x \right)} = 3$.

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

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

      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 $1$

         $g{\left(x \right)} = \sin{\left(\frac{x}{3} \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

         1. Let $u = \frac{x}{3}$.

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

            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: $\frac{1}{3}$

            The result of the chain rule is:

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

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

      So, the result is: $a \left(\frac{x \cos{\left(\frac{x}{3} \right)}}{3} + \sin{\left(\frac{x}{3} \right)}\right)$

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

   1. The derivative of the constant $3$ is zero.

   Now plug in to the quotient rule:

   $\frac{a \left(\frac{x \cos{\left(\frac{x}{3} \right)}}{3} + \sin{\left(\frac{x}{3} \right)}\right)}{3}$

2. Now simplify:

   $$\frac{a \left(x \cos{\left(\frac{x}{3} \right)} + 3 \sin{\left(\frac{x}{3} \right)}\right)}{9}$$

The answer is:

$$\frac{a \left(x \cos{\left(\frac{x}{3} \right)} + 3 \sin{\left(\frac{x}{3} \right)}\right)}{9}$$