Derivative of y=(x⁴-x²+1)³

1. Let $u = \left(x^{4} - x^{2}\right) + 1$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{4} - x^{2}\right) + 1\right)$:

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

      1. Differentiate $x^{4} - x^{2}$ 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: $x^{2}$ goes to $2 x$

            So, the result is: $- 2 x$

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

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

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

   The result of the chain rule is:

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

4. Now simplify:

   $$6 x \left(2 x^{2} - 1\right) \left(x^{4} - x^{2} + 1\right)^{2}$$

The answer is: