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