The derivative of sin(tan^-1e^2x) with respect to x is:

$y=\sin(\tan^{-1}e^{2x})$ differentiating w.r.t x

$\frac{dy}{dx}=\cos(\tan^{-1}e^{2x})\frac{d}{dx}(\tan^{-1}e^{2x})⇒\frac{dy}{dx}=\frac{\cos(\tan^{-1}e^{2x})}{1+e^{4x}}\frac{de^{2x}}{x}$

$=\frac{2e^{2x}\cos(\tan^{-1}e^{2x})}{1+e^{4x}}$