Derivative of ln((x+1)*(x+2))+3ln((x+2))−3ln((x+1))2

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

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

      1. Let $u = \left(x + 1\right) \left(x + 2\right)$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right) \left(x + 2\right)$:

         1. Apply the product rule:

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

            $f{\left(x \right)} = x + 1$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

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

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

               The result is: $1$

            $g{\left(x \right)} = x + 2$; to find $\frac{d}{d x} g{\left(x \right)}$:

            1. Differentiate $x + 2$ term by term:

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

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

               The result is: $1$

            The result is: $2 x + 3$

         The result of the chain rule is:

         $$\frac{2 x + 3}{\left(x + 1\right) \left(x + 2\right)}$$

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

         1. Let $u = x + 2$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 2\right)$:

            1. Differentiate $x + 2$ term by term:

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

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

               The result is: $1$

            The result of the chain rule is:

            $$\frac{1}{x + 2}$$

         So, the result is: $\frac{3}{x + 2}$

      The result is: $\frac{3}{x + 2} + \frac{2 x + 3}{\left(x + 1\right) \left(x + 2\right)}$

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

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

         1. Let $u = x + 1$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$:

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

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

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

               The result is: $1$

            The result of the chain rule is:

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

         So, the result is: $\frac{3}{x + 1}$

      So, the result is: $- \frac{6}{x + 1}$

   The result is: $\frac{3}{x + 2} - \frac{6}{x + 1} + \frac{2 x + 3}{\left(x + 1\right) \left(x + 2\right)}$

2. Now simplify:

   $$- \frac{x + 6}{x^{2} + 3 x + 2}$$

The answer is:

$$- \frac{x + 6}{x^{2} + 3 x + 2}$$