Derivative of y=(x³+2x)(2x-1)

1. Apply the product rule:

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

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

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

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

      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: $2$

      The result is: $3 x^{2} + 2$

   $g{\left(x \right)} = 2 x - 1$; 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: $2 x^{3} + 4 x + \left(2 x - 1\right) \left(3 x^{2} + 2\right)$

2. Now simplify:

   $$8 x^{3} - 3 x^{2} + 8 x - 2$$

The answer is:

$$8 x^{3} - 3 x^{2} + 8 x - 2$$