Derivative of (5x-9x^3)(8+x^2)

1. Apply the product rule:

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

   $f{\left(x \right)} = - 9 x^{3} + 5 x$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $- 9 x^{3} + 5 x$ 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: $5$

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         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: $27 x^{2}$

         So, the result is: $- 27 x^{2}$

      The result is: $5 - 27 x^{2}$

   $g{\left(x \right)} = x^{2} + 8$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $x^{2} + 8$ term by term:

      1. The derivative of the constant $8$ is zero.

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

      The result is: $2 x$

   The result is: $2 x \left(- 9 x^{3} + 5 x\right) + \left(5 - 27 x^{2}\right) \left(x^{2} + 8\right)$

2. Now simplify:

   $$- 45 x^{4} - 201 x^{2} + 40$$

The answer is:

$$- 45 x^{4} - 201 x^{2} + 40$$