Function Expression :
\[f(x)=\frac{2x}{x+1}-x \]Domain
\[\left]-\infty, -1\right[ \cup \left]-1, \infty\right[ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = +\infty \]\[\lim_{x \overset{<}{\rightarrow-1} }f(x) = +\infty \]
\[\lim_{x \overset{>}{\rightarrow-1} }f(x) = -\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = -\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=- \frac{2 x}{\left(x + 1\right)^{2}} - 1 + \frac{2}{x + 1} \]\[f^{\,\prime}(x)=-1 + \frac{2}{\left(x + 1\right)^{2}} \]
\[f^{\,\prime}(x)=\frac{2 - \left(x + 1\right)^{2}}{\left(x + 1\right)^{2}} \]
Integral
\[F(x) = - \frac{x^{2}}{2} + 2 x - 2 \log{\left(x + 1 \right)} \]Sign Table
Variation Table
Plot
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