Differentiate the logarithmic series to obtain a logistic expansion

$y^{\prime}=e^x/(1+e^x)$. Differentiation gives $y^{\prime\prime}=e^x/(1+e^x)^2=e^x/(1+e^x)-[e^x/(1+e^x)]^2=y^{\prime}-(y^{\prime})^2$.

$y(0)=\ln2$, $y^{\prime}(0)=1/2$ and $y^{\prime\prime}(0)=1/4$. Thus $y=\ln2+x/2+x^2/8+O(x^4)$ locally. Its cubic coefficient is zero: differentiating the identity gives $y^{\prime\prime\prime}=y^{\prime\prime}(1-2y^{\prime})$, which vanishes at zero.

Termwise differentiation yields $e^x/(1+e^x)=1/2+x/4+O(x^3)$. The source’s final line prints the derivative of $\ln(1+e^{2x})$; that exponent is a typo. Its intended derivative is $d[\ln(1+e^x)]/dx$, consistent with its earlier identity and final coefficients.