Derivative of y=(e^(3x))/3sin(x)

1. Apply the quotient rule, which is:

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

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

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

   1. Apply the product rule:

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

      $f{\left(x \right)} = e^{3 x}$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. Let $u = 3 x$.

      2. The derivative of $e^{u}$ is itself.

      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:

         $$3 e^{3 x}$$

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

      1. The derivative of sine is cosine:

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

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

   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:

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

2. Now simplify:

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

The answer is:

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