Derivative of 1/sqrt(1+(x-1)^4)

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

2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

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

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

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

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

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

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

         2. Let $u = x - 1$.

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

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

            1. Differentiate $x - 1$ term by term:

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

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

               The result is: $1$

            The result of the chain rule is:

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

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

      The result of the chain rule is:

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

   The result of the chain rule is:

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

4. Now simplify:

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

The answer is: