First of all, I am assuming that there is a mistake in the formatting, and that you mean e^{2x} instead of e^2x.
This is a composite function. Which means, a function f of the form f(x)=g(h(x)). In your case, g(x)=\arctan(x), and h(x)=e^{2x}. To derive such functions, you must use the chain rule, which states that, using the same notation I introduced above, f'(x)=g'(h(x))*h'(x). The two derivatives we need are both elementary, so I will not proof the results:
d/{dx} \arctan(x)=1/{x^2+1}
d/dx e^{2x}=2e^{2x}
Note that for the second derivative, we actually use the chain rule one more time, because e^{2x} is of the form a(b(x), with a(x)=e^x, and b(x)=2x.
The final answer is thus, as said above,
g'(h(x))*h'(x)=1/{e^{4x}+1}*2e^{2x}={2e^{2x}}/{e^{4x}+1}
