Differentiate the following functions w.r.t. x. : (2 e^x-1 ) (2 e… - Vidyadip

Question

Differentiate the following functions w.r.t. x. : $\frac{\left(2 e^x-1\right)}{\left(2 e^x+1\right)}$

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Answer

Let $y=\frac{2 \mathrm{e}^x-1}{2 \mathrm{e}^x+1}$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
& \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{2 \mathrm{e}^x-1}{2 \mathrm{e}^x+1}\right) \text { } \\
& =\frac{\left(2 \mathrm{e}^x+1\right) \frac{\mathrm{d}}{\mathrm{d} x}\left(2 \mathrm{e}^x-1\right)-\left(2 \mathrm{e}^x-1\right) \frac{\mathrm{d}}{\mathrm{d} x}\left(2 \mathrm{e}^x+1\right)}{\left(2 \mathrm{e}^x+1\right)^2} \\
& =\frac{\left(2 \mathrm{e}^x+1\right)\left(2 \mathrm{e}^x-0\right)-\left(2 \mathrm{e}^x-1\right)\left(2 \mathrm{e}^x+0\right)}{\left(2 \mathrm{e}^x+1\right)^2} \\
& =\frac{\left(2 \mathrm{e}^x+1\right)\left(2 \mathrm{e}^x\right)-\left(2 \mathrm{e}^x-1\right)\left(2 \mathrm{e}^x\right)}{\left(2 \mathrm{e}^x+1\right)^2} \\
& =\frac{2 \mathrm{e}^x\left(2 \mathrm{e}^x+1-2 \mathrm{e}^x+1\right)}{\left(2 \mathrm{e}^x+1\right)^2} \\
& =\frac{2 \mathrm{e}^x(2)}{\left(2 \mathrm{e}^x+1\right)^2} \\
& =\frac{4 \mathrm{e}^x}{\left(2 \mathrm{e}^x+1\right)^2}
\end{aligned}
$

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