Derivative of y=x^2(2x+1)

1. Apply the product rule:

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

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

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

   $g{\left(x \right)} = 2 x + 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $2 x + 1$ 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 $1$ is zero.

      The result is: $2$

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

2. Now simplify:

   $$2 x \left(3 x + 1\right)$$

The answer is:

$$2 x \left(3 x + 1\right)$$