$$\frac{a^{-1}-b^{-1}}{a^{-1}+b^{-1}}$$
$\frac{b-a}{a+b}$
$$\frac{\left(-\frac{1}{b}a+1\right)\times \frac{1}{a}}{\left(1+\frac{1}{b}a\right)\times \frac{1}{a}}$$
$$\frac{-\frac{1}{b}a+1}{1+\frac{1}{b}a}$$
$$\frac{-\frac{a}{b}+1}{1+\frac{1}{b}a}$$
$$\frac{-\frac{a}{b}+\frac{b}{b}}{1+\frac{1}{b}a}$$
$$\frac{\frac{-a+b}{b}}{1+\frac{1}{b}a}$$
$$\frac{\frac{-a+b}{b}}{1+\frac{a}{b}}$$
$$\frac{\frac{-a+b}{b}}{\frac{b}{b}+\frac{a}{b}}$$
$$\frac{\frac{-a+b}{b}}{\frac{b+a}{b}}$$
$$\frac{\left(-a+b\right)b}{b\left(b+a\right)}$$
$$\frac{-a+b}{a+b}$$
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