Function Expression :
\[f(x)=\frac{e^{2x}}{2e^{2x}-2e^x+1} \]Domain
\[]-\infty ;+\infty [ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = 0 \]\[\lim_{x \rightarrow+\infty}f(x) = \frac{1}{2} \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{\left(- 4 e^{2 x} + 2 e^{x}\right) e^{2 x}}{\left(2 e^{2 x} - 2 e^{x} + 1\right)^{2}} + \frac{2 e^{2 x}}{2 e^{2 x} - 2 e^{x} + 1} \]\[f^{\,\prime}(x)=\frac{2 \cdot \left(1 - e^{x}\right) e^{2 x}}{4 e^{4 x} - 8 e^{3 x} + 8 e^{2 x} - 4 e^{x} + 1} \]
\[ \]
Integral
\[F(x) = \operatorname{RootSum} {\left(8 z^{2} - 4 z + 1, \left( i \mapsto i \log{\left(2 i + e^{x} - 1 \right)} \right)\right)} \]Sign Table
Variation Table
Plot
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