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