Derivative of (x^2-2)*(x^7+4)

1. Apply the product rule:

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

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

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

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

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

      The result is: $2 x$

   $g{\left(x \right)} = x^{7} + 4$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $x^{7} + 4$ term by term:

      1. Apply the power rule: $x^{7}$ goes to $7 x^{6}$

      2. The derivative of the constant $4$ is zero.

      The result is: $7 x^{6}$

   The result is: $7 x^{6} \left(x^{2} - 2\right) + 2 x \left(x^{7} + 4\right)$

2. Now simplify:

   $$x \left(9 x^{7} - 14 x^{5} + 8\right)$$

The answer is:

$$x \left(9 x^{7} - 14 x^{5} + 8\right)$$