Derivative of y=ln(x²+3x+2)

1. Let $u = \left(x^{2} + 3 x\right) + 2$.

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

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

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

      1. Differentiate $x^{2} + 3 x$ term by term:

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

         2. 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: $3$

         The result is: $2 x + 3$

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

      The result is: $2 x + 3$

   The result of the chain rule is:

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

4. Now simplify:

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

The answer is: