`f(x) = arctan(e^x)` Find the derivative of the function

The given function: `f(x) =arctan(e^x)`  is in a form of inverse trigonometric function.

It can be evaluated using the derivative formula for inverse of tangent function:

`d/(dx)arctan(u) = ((du)/(dx))/(1+x^2)` .

 We  let `u = e^x` then  `(du)/(dx)= d/(dx) (e^x)= e^x` .

Applying the the formula, we get:

`f'(x)= d/(dx) arctan(e^x)`

          `=e^x/(1 +(e^x)^2)`

Using the law of exponent: `(x^n)^m=x^(n*m)` , we may simplify the part:

`(e^x)^2 = e^((x*2)) = e^(2x)`

The derivative of the function` f(x) = arctan(e^x)` becomes:

`f'(x)= e^x/(1 +e^(2x)) ` as the Final Answer.