Derivative of 4ln3x+lnx

1. Differentiate $\log{\left(x \right)} + 4 \log{\left(3 x \right)}$ term by term:

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

      1. 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: $\frac{4}{x}$

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

   The result is: $\frac{5}{x}$

The answer is:

$$\frac{5}{x}$$