Derivative of 5x-ln(5x)+12

1. Differentiate $\left(5 x - \log{\left(5 x \right)}\right) + 12$ term by term:

   1. Differentiate $5 x - \log{\left(5 x \right)}$ term by term:

      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: $5$

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

         1. Let $u = 5 x$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

            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: $5$

            The result of the chain rule is:

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

         So, the result is: $- \frac{1}{x}$

      The result is: $5 - \frac{1}{x}$

   2. The derivative of the constant $12$ is zero.

   The result is: $5 - \frac{1}{x}$

The answer is:

$$5 - \frac{1}{x}$$