Derivative of log(4)(2x/x+1)

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

   1. Differentiate $1 + \frac{2 x}{x}$ term by term:

      1. Apply the quotient rule, which is:

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

         $f{\left(x \right)} = 2 x$ and $g{\left(x \right)} = x$.

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

         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: $2$

         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:

         $0$

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

      The result is: $0$

   So, the result is: $0$

The answer is:

$$0$$