$$\frac{a{b}^{-1}-a^{-1}b}{a^{-2}-2a^{-1}{b}^{-1}+b^{-2}}$$

$$=\frac{a·{\displaystyle \frac{1}{b^1}}-{\displaystyle \frac{1}{a^1}}·b}{{\displaystyle \frac{1}{a^2}}-2·{\displaystyle \frac{1}{a^1}}·{\displaystyle \frac{1}{b^1}}+{\displaystyle \frac{1}{b^2}}}$$

$$=\frac{{\displaystyle \frac{a}{1}}·{\displaystyle \frac{1}{b}}-{\displaystyle \frac{1}{a}}·{\displaystyle \frac{b}{1}}}{{\displaystyle \frac{1}{a^2}}-{\displaystyle \frac{2}{1}}·{\displaystyle \frac{1}{a}}·{\displaystyle \frac{1}{b}}+{\displaystyle \frac{1}{b^2}}}$$

$$=\frac{{\displaystyle \frac{a}{b}-\frac{b}{a}}}{{\displaystyle \frac{1}{a^2}}-{\displaystyle \frac{2}{ab}}+{\displaystyle \frac{1}{b^2}}}$$

$$=\frac{{\displaystyle \frac{a}{b}·\frac{a}{a}-\frac{b}{a}·\frac{b}{b}}}{{\displaystyle \frac{1}{a^2}·}{\displaystyle \frac{b^2}{b^2}}-{\displaystyle \frac{2}{ab}·\frac{ab}{ab}}+{\displaystyle \frac{1}{b^2}·\frac{a^2}{a^2}}}$$

$$=\frac{{\displaystyle \frac{a^2}{ab}-\frac{b^2}{ab}}}{{\displaystyle \frac{b^2}{a^2{b}^2}}-{\displaystyle \frac{2ab}{a^2{b}^2}}+{\displaystyle \frac{a^2}{a^2{b}^2}}}$$

$$=\frac{{\displaystyle \frac{a^2-b^2}{ab}}}{{\displaystyle \frac{b^2-2ab+a^2}{a^2{b}^2}}}$$

$$=\frac{a^2-b^2}{ab}÷\frac{b^2-2ab+a^2}{a^2{b}^2}$$

$$=\frac{a^2-b^2}{\cancel{ab}}·\frac{a^{\cancel{2}}{b}^{\cancel{2}}}{b^2-2ab+a^2}$$

$$=\frac{a^2-b^2}{1}·\frac{ab}{{\left(a-b\right)}^2}$$

$$=\frac{\bcancel{\left(a-b\right)}\left(a+b\right)}{1}·\frac{ab}{\bcancel{\left(a-b\right)}\left(a-b\right)}$$