Derivative of 14x-ln(14x)+8

1. Differentiate $\left(14 x - \log{\left(14 x \right)}\right) + 8$ term by term:

   1. Differentiate $14 x - \log{\left(14 x \right)}$ 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: $14$

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

         1. Let $u = 14 x$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 14 x$:

            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: $14$

            The result of the chain rule is:

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

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

      The result is: $14 - \frac{1}{x}$

   2. The derivative of the constant $8$ is zero.

   The result is: $14 - \frac{1}{x}$

The answer is:

$$14 - \frac{1}{x}$$