Function Expression :
\[f(x)=\sqrt{e^x-e^{2x}} \]Domain
\[\left]-\infty, 0\right] \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = 0 \]\[\lim_{x \overset{<}{\rightarrow0} }f(x) = 0 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{- e^{2 x} + \frac{e^{x}}{2}}{\sqrt{- e^{2 x} + e^{x}}} \]\[f^{\,\prime}(x)=\frac{\left(\frac{1}{2} - e^{x}\right) e^{x}}{\sqrt{\left(1 - e^{x}\right) e^{x}}} \]
\[f^{\,\prime}(x)=\frac{\left(1 - 2 e^{x}\right) e^{x}}{2 \sqrt{\left(1 - e^{x}\right) e^{x}}} \]
Integral
\[F(x) = \int \sqrt{- e^{2 x} + e^{x}}\, dx \]Sign Table
Variation Table
Plot
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