$\therefore \frac{{dy}}{{dx}} = \frac{{({e^{2x}} - {e^{ - 2x}})2({e^{2x}} - {e^{ - 2x}}) - ({e^{2x}} + {e^{ - 2x}})2({e^{2x}} + {e^{ - 2x}})}}{{{{({e^{2x}} - {e^{ - 2x}})}^2}}}$
$ = \frac{{ - 8}}{{{{({e^{2x}} - {e^{ - 2x}})}^2}}}$.
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$S :$ Both $\sin x$ and cosx are decreasing functions in $\left( {{\pi \over 2},\pi } \right)$
$R:$ If a differentiable function decreases in $(a, b)$ then its derivative also decreases in $ (a, b).$
Which of the following is true
$( S 1): \lim _{ n \rightarrow \infty} \frac{1}{ n ^2}(2+4+6+\ldots \ldots \ldots+2 n)=1$
(S2) : $\lim _{ n \rightarrow \infty} \frac{1}{ n ^{16}}\left(1^{15}+2^{15}+3^{15}+\ldots \ldots \ldots .+ n ^{15}\right)=\frac{1}{16}$
$I$. $\left(\sum_{k=1}^{2018} k x_k\right)^2 \leq N\left(\sum_{k=1}^{2018} k x_k^2\right)$
$II$. $\left(\sum_{k=1}^{2018} k x_k\right)^2 \leq N\left(\sum_{k=1}^{2018} k^2 x_k^2\right)$ Then,