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

1. Let $u = \left(x^{4} - x\right) - 1$.

2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{4} - x\right) - 1\right)$:

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

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

         The result is: $4 x^{3} - 1$

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

      The result is: $4 x^{3} - 1$

   The result of the chain rule is:

   $$4 \left(4 x^{3} - 1\right) \left(\left(x^{4} - x\right) - 1\right)^{3}$$

4. Now simplify:

   $$\left(4 - 16 x^{3}\right) \left(- x^{4} + x + 1\right)^{3}$$

The answer is:

$$\left(4 - 16 x^{3}\right) \left(- x^{4} + x + 1\right)^{3}$$