Differentiate the following functions w.r.t. x. : (x+1)(x-1) (e^x… - Vidyadip

Question

Differentiate the following functions w.r.t. x. : $\frac{(x+1)(x-1)}{\left(e^x+1\right)}$

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Answer

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

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