Find the derivative of the function. \( f(x)=\arctan \left(e^{x}\right) \) \( f^{\prime}(x)=\square \)

To find the derivative of the function \( f(x) = \arctan(e^{x}) \), we can apply the chain rule. The derivative of \( \arctan(u) \) is \( \frac{1}{1+u^2} \), where \( u = e^{x} \). Thus, the derivative of \( f(x) \) is: \[ f'(x) = \frac{1}{1 + (e^{x})^2} \cdot \frac{d}{dx}(e^{x}) = \frac{1}{1 + e^{2x}} \cdot e^{x} \] So, we have: \[ f'(x) = \frac{e^{x}}{1 + e^{2x}} \]