Derivative of log2(x)+3log3x

1. Differentiate $\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 3 \log{\left(3 x \right)}$ term by term:

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

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

      So, the result is: $\frac{1}{x \log{\left(2 \right)}}$

   2. 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 $\log{\left(u \right)}$ is $\frac{1}{u}$.

      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:

         $$\frac{1}{x}$$

      So, the result is: $\frac{3}{x}$

   The result is: $\frac{1}{x \log{\left(2 \right)}} + \frac{3}{x}$

2. Now simplify:

   $$\frac{1 + \log{\left(8 \right)}}{x \log{\left(2 \right)}}$$

The answer is: