Derivative of x-1-x/x²+1

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

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

      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: $1$

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

         1. Apply the quotient rule, which is:

            $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

            $f{\left(x \right)} = x$ and $g{\left(x \right)} = x^{2}$.

            To find $\frac{d}{d x} f{\left(x \right)}$:

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

            To find $\frac{d}{d x} g{\left(x \right)}$:

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

            Now plug in to the quotient rule:

            $- \frac{1}{x^{2}}$

         So, the result is: $\frac{1}{x^{2}}$

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

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

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

The answer is:

$$1 + \frac{1}{x^{2}}$$