证明: 函数 $y=\frac{x}{x+1}$ 在 $(-1,+\infty)$ 上是增函数.

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设 $x_{1}>x_{2}>-1$, 则

$y_{1}-y_{2}=\frac{x_{1}}{x_{1}+1}-\frac{x_{2}}{x_{2}+1}=\frac{x_{1}-x_{2}}{x_{1}+1 \quad x_{2}+1}$.

$\because x_{1}>x_{2}>-1, \quad \therefore x_{1}-x_{2}>0, \quad x_{1}+1>0, x_{2}+1>0$,

$\therefore \frac{x_{1}-x_{2}}{x_{1}+1 x_{2}+1}>0$, 即 $y_{1}-y_{2}>0, y_{1}>y_{2}$,

$\therefore y=\frac{x}{x+1}$ 在 $(-1,+\infty)$ 上是增函数.