Differentiate the following functions with respect to x :

$\frac{e^{x}}{1 + e^{x}}$

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Solution

we have

$\frac{d}{d x} (\frac{e^{x}}{1 + e^{x}})$

Using quotient rule,

$= \frac{(1 + x^{2}) \frac{d}{d x} (e^{x}) - (e^{x}) \frac{d}{d x} (1 + x^{2})}{(1 + x^{2})^{2}} = \frac{(1 + x^{2}) e^{x} - e^{x} \times 2 x}{(1 + x^{2})^{2}} = \frac{e^{x} (1 + x^{2} - 2 x)}{(1 + x^{2})^{2}} = \frac{e^{x} (1 - x)^{2}}{(1 + x^{2})^{2}}$

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