Derivative of y=x³ln(x²+4x)

1. Apply the product rule:

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

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

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

   $g{\left(x \right)} = \log{\left(x^{2} + 4 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = x^{2} + 4 x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 4 x\right)$:

      1. Differentiate $x^{2} + 4 x$ 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: $4$

         The result is: $2 x + 4$

      The result of the chain rule is:

      $$\frac{2 x + 4}{x^{2} + 4 x}$$

   The result is: $\frac{x^{3} \cdot \left(2 x + 4\right)}{x^{2} + 4 x} + 3 x^{2} \log{\left(x^{2} + 4 x \right)}$

2. Now simplify:

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

The answer is: