Derivative of y=(2x^3-3)(3x^2-2)

1. Apply the product rule:

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

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

   1. Differentiate $2 x^{3} - 3$ 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: $x^{3}$ goes to $3 x^{2}$

         So, the result is: $6 x^{2}$

      2. The derivative of the constant $\left(-1\right) 3$ is zero.

      The result is: $6 x^{2}$

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

   1. Differentiate $3 x^{2} - 2$ 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: $x^{2}$ goes to $2 x$

         So, the result is: $6 x$

      2. The derivative of the constant $\left(-1\right) 2$ is zero.

      The result is: $6 x$

   The result is: $6 x^{2} \cdot \left(3 x^{2} - 2\right) + 6 x \left(2 x^{3} - 3\right)$

2. Now simplify:

   $$6 x \left(5 x^{3} - 2 x - 3\right)$$

The answer is:

$$6 x \left(5 x^{3} - 2 x - 3\right)$$