Derivative of (log^3)(2x+3)^2

1. Apply the product rule:

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

   $f{\left(x \right)} = \log{\left(x \right)}^{3}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = \log{\left(x \right)}$.

   2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

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

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

      The result of the chain rule is:

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

   $g{\left(x \right)} = \left(2 x + 3\right)^{2}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 2 x + 3$.

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

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

      1. Differentiate $2 x + 3$ term by term:

         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: $2$

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

         The result is: $2$

      The result of the chain rule is:

      $$8 x + 12$$

   The result is: $\left(8 x + 12\right) \log{\left(x \right)}^{3} + \frac{3 \left(2 x + 3\right)^{2} \log{\left(x \right)}^{2}}{x}$

2. Now simplify:

   $$\frac{\left(2 x + 3\right) \left(4 x \log{\left(x \right)} + 6 x + 9\right) \log{\left(x \right)}^{2}}{x}$$

The answer is: