Derivative of (x^2-x-1)^8

1. Let $u = \left(x^{2} - x\right) - 1$.

2. Apply the power rule: $u^{8}$ goes to $8 u^{7}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{2} - x\right) - 1\right)$:

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

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

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

         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: $-1$

         The result is: $2 x - 1$

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

      The result is: $2 x - 1$

   The result of the chain rule is:

   $$8 \left(2 x - 1\right) \left(\left(x^{2} - x\right) - 1\right)^{7}$$

4. Now simplify:

   $$\left(8 - 16 x\right) \left(- x^{2} + x + 1\right)^{7}$$

The answer is:

$$\left(8 - 16 x\right) \left(- x^{2} + x + 1\right)^{7}$$