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

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

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

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

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

      1. Let $u = x - 1$.

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

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

         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:

         $$2 x - 2$$

      The result of the chain rule is:

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

   So, the result is: $\frac{2 x - 2}{\left(x - 1\right)^{4}}$

2. Now simplify:

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

The answer is:

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