Given,

$f\left(x\right)={\log }_e\left(e^{2x}{\left(\frac{3x+5}{5-3x}\right)}^{\frac{2}{3}}\right)$

$\Rightarrow f\left(x\right)={\log }_e{e}^{2x}+{\log }_e{\left(\frac{3x+5}{5-3x}\right)}^{\frac{2}{3}}$

$\Rightarrow f\left(x\right)=2x+\frac{2}{3}{\log }_e\left(\frac{3x+5}{5-3x}\right)$

$\Rightarrow f\left(x\right)=2x+\frac{2}{3}\left({\log }_e\left(3x+5\right)-{\log }_e\left(5-3x\right)\right)$

Now differentiating the function with respect to $x$ we get,

$\frac{df}{dx}=2+\frac{2}{3}\left(\frac{1}{3x+5}\times 3-\frac{1}{5-3x}\times \left(-3\right)\right)$

$\Rightarrow \frac{df}{dx}=2+2\left(\frac{1}{3x+5}+\frac{1}{5-3x}\right)$

$\Rightarrow {\left(\frac{df}{dx}\right)}_{x=1}=2+2\left(\frac{1}{3+5}+\frac{1}{5-3}\right)=2+2\times \frac{5}{8}=\frac{13}{4}$