# How do you find the derivative of g(x) = −2/(x + 1) using the limit definition?

Aug 13, 2016

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

#### Explanation:

$f ' \left(x\right) = {\lim}_{h \rightarrow 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

$= {\lim}_{h \rightarrow 0} \frac{- \frac{2}{x + h + 1} + \frac{2}{x + 1}}{h}$

$= {\lim}_{h \rightarrow 0} \frac{\frac{- 2 \left(x + 1\right)}{\left(x + h + 1\right) \left(x + 1\right)} + \frac{2 \left(x + h + 1\right)}{\left(x + h + 1\right) \left(x + 1\right)}}{h}$

$= {\lim}_{h \rightarrow 0} \frac{\frac{2 h}{\left(x + h + 1\right) \left(x + 1\right)}}{h} = {\lim}_{h \rightarrow 0} \frac{2}{\left(x + h + 1\right) \left(x + 1\right)}$

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