Differentiate the following functions w.r.t. x. : 1 e^x+1

Question

Differentiate the following functions w.r.t. x. : \begin{equation}
\frac{1}{e^x+1}
\end{equation}

✓

Answer

Let $y=\frac{1}{\mathrm{e}^x+1}$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{1}{\mathrm{e}^x+1}\right) \\
& =\frac{\left(\mathrm{e}^x+1\right) \frac{\mathrm{d}}{\mathrm{d} x}(1)-(1) \frac{\mathrm{d}}{\mathrm{d} x}\left(\mathrm{e}^x+1\right)}{\left(\mathrm{e}^x+1\right)^2} \\
& =\frac{\left(\mathrm{e}^x+1\right)(0)-(1)\left(\mathrm{e}^x+0\right)}{\left(\mathrm{e}^x+1\right)^2} \\
& =\frac{\mathrm{e}^x+1-\mathrm{e}^x}{\left(\mathrm{e}^x+1\right)^2} \\
& =\frac{1}{\left(\mathrm{e}^x+1\right)^2}
\end{aligned}
$

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