Derivative of x(x^4-2x-1)

1. Apply the product rule:

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

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

   1. Apply the power rule: $x$ goes to $1$

   $g{\left(x \right)} = \left(x^{4} - 2 x\right) - 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $\left(x^{4} - 2 x\right) - 1$ term by term:

      1. Differentiate $x^{4} - 2 x$ term by term:

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

         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: $4 x^{3} - 2$

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

      The result is: $4 x^{3} - 2$

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

2. Now simplify:

   $$5 x^{4} - 4 x - 1$$

The answer is:

$$5 x^{4} - 4 x - 1$$