Derivative of y=(x²+3)(x⁴-1)

1. Apply the product rule:

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

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

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

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

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

      The result is: $2 x$

   $g{\left(x \right)} = x^{4} - 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $x^{4} - 1$ term by term:

      1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$

      2. The derivative of the constant $-1$ is zero.

      The result is: $4 x^{3}$

   The result is: $4 x^{3} \left(x^{2} + 3\right) + 2 x \left(x^{4} - 1\right)$

2. Now simplify:

   $$6 x^{5} + 12 x^{3} - 2 x$$

The answer is:

$$6 x^{5} + 12 x^{3} - 2 x$$