Derivative of log3(x)*sin2x

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = \log{\left(x \right)} \sin{\left(2 x \right)}$ and $g{\left(x \right)} = \log{\left(3 \right)}$.

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

   1. Apply the product rule:

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

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

      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$.

      $g{\left(x \right)} = \sin{\left(2 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

      1. Let $u = 2 x$.

      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 $\frac{d}{d x} 2 x$:

         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: $2$

         The result of the chain rule is:

         $$2 \cos{\left(2 x \right)}$$

      The result is: $2 \log{\left(x \right)} \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{x}$

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

   1. The derivative of the constant $\log{\left(3 \right)}$ is zero.

   Now plug in to the quotient rule:

   $\frac{2 \log{\left(x \right)} \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{x}}{\log{\left(3 \right)}}$

2. Now simplify:

   $$\frac{2 x \log{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)}}{x \log{\left(3 \right)}}$$

The answer is: