The derivative of \(\displaystyle\frac{e^{x}}{1 + e^{x}}\) with respect to x is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{e^{x}}{(1 + e^{x})^{2}}\)
By the quotient rule, with numerator \(e^{x}\) and denominator \(1 + e^{x}\):
\( \displaystyle \frac{d}{dx}\left(\frac{e^{x}}{1 + e^{x}}\right) = \frac{(1 + e^{x})e^{x} - e^{x} \cdot e^{x}}{(1 + e^{x})^{2}} \)
\( \displaystyle = \frac{e^{x} + e^{2x} - e^{2x}}{(1 + e^{x})^{2}} = \frac{e^{x}}{(1 + e^{x})^{2}} \)