Function Expression :
\[f(x)=\frac{1}{1+e^x}-x \]Domain
\[]-\infty ;+\infty [ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = +\infty \]\[\lim_{x \rightarrow+\infty}f(x) = -\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=-1 - \frac{e^{x}}{\left(e^{x} + 1\right)^{2}} \]\[f^{\,\prime}(x)=-1 - \frac{1}{4 \cosh^{2}{\left(\frac{x}{2} \right)}} \]
\[f^{\,\prime}(x)=\frac{- 4 \cosh^{2}{\left(\frac{x}{2} \right)} - 1}{4 \cosh^{2}{\left(\frac{x}{2} \right)}} \]
Integral
\[F(x) = - \frac{x^{2}}{2} + x - \log{\left(e^{x} + 1 \right)} \]Sign Table
Variation Table
Plot
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