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

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

   1. Differentiate $x + \frac{4}{x - 1}$ term by term:

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

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

         1. Let $u = x - 1$.

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

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

            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:

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

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

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

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

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

2. Now simplify:

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

The answer is:

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