If $e^{g\left(y\right)}-e^{-g\left(y\right)}=2f\left(x\right)\text{ then }\frac{dy}{dx}=$

$e^{g\left(y\right)}-e^{-g\left(y\right)}=2f\left(x\right)$

$e^{g\left(y\right)}{g}^1\left(y\right)\frac{dy}{dx}+e^{-g\left(y\right)}{g}^1\left(y\right)\frac{dy}{dx}=2f^1\left(x\right)$

$\frac{dy}{dx}\left(e^{g\left(y\right)}+e^{-g\left(y\right)}\right)g\text{'}\left(y\right)=2f\text{'}\left(x\right)$

$\frac{dy}{dx}g\text{'}\left(y\right)\sqrt{{\left(e^{g\left(y\right)}-e^{-g\left(y\right)}\right)}^2+4}=2f\text{'}\left(x\right)$

$\frac{dy}{dx}=\frac{2f\text{'}\left(x\right)}{g\text{'}\left(y\right)\sqrt{{\left(2f\left(x\right)\right)}^2+4}}=\frac{f\text{'}\left(x\right)}{g\text{'}\left(y\right)\sqrt{\left(f{\left(x\right)}^2\right)+1}}$