Derivative of y=(√x-1)²-(x²+1)⁴

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

   1. Let $u = \sqrt{x} - 1$.

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

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

      1. Differentiate $\sqrt{x} - 1$ term by term:

         1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

         2. The derivative of the constant $\left(-1\right) 1$ is zero.

         The result is: $\frac{1}{2 \sqrt{x}}$

      The result of the chain rule is:

      $$\frac{2 \sqrt{x} - 2}{2 \sqrt{x}}$$

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

      1. Let $u = x^{2} + 1$.

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

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

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

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

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

            The result is: $2 x$

         The result of the chain rule is:

         $$8 x \left(x^{2} + 1\right)^{3}$$

      So, the result is: $- 8 x \left(x^{2} + 1\right)^{3}$

   The result is: $- 8 x \left(x^{2} + 1\right)^{3} + \frac{2 \sqrt{x} - 2}{2 \sqrt{x}}$

2. Now simplify:

   $$- 8 x \left(x^{2} + 1\right)^{3} + 1 - \frac{1}{\sqrt{x}}$$

The answer is: