Derivative of 4-(4-lnx)/x

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

   1. The derivative of the constant $4$ is zero.

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

      1. Apply the quotient rule, which is:

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

         $f{\left(x \right)} = 4 - \log{\left(x \right)}$ and $g{\left(x \right)} = x$.

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

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

            1. The derivative of the constant $4$ is zero.

            2. 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: $- \frac{1}{x}$

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

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

         1. Apply the power rule: $x$ goes to $1$

         Now plug in to the quotient rule:

         $\frac{\log{\left(x \right)} - 5}{x^{2}}$

      So, the result is: $- \frac{\log{\left(x \right)} - 5}{x^{2}}$

   The result is: $- \frac{\log{\left(x \right)} - 5}{x^{2}}$

2. Now simplify:

   $$\frac{5 - \log{\left(x \right)}}{x^{2}}$$

The answer is: