Derivative of y=2ln((x+3)/x)-3

1. Differentiate $2 \log{\left(\frac{x + 3}{x} \right)} - 3$ term by term:

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

      1. Let $u = \frac{x + 3}{x}$.

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

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

         1. Apply the quotient rule, which is:

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

            $f{\left(x \right)} = x + 3$ and $g{\left(x \right)} = x$.

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

            1. Differentiate $x + 3$ term by term:

               1. The derivative of the constant $3$ is zero.

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

               The result is: $1$

            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:

            $- \frac{3}{x^{2}}$

         The result of the chain rule is:

         $$- \frac{3}{x \left(x + 3\right)}$$

      So, the result is: $- \frac{6}{x \left(x + 3\right)}$

   2. The derivative of the constant $-3$ is zero.

   The result is: $- \frac{6}{x \left(x + 3\right)}$

2. Now simplify:

   $$- \frac{6}{x \left(x + 3\right)}$$

The answer is:

$$- \frac{6}{x \left(x + 3\right)}$$