Calculate the derivative of the following function. \[ y=\tan \left(e^{x}\right) \]

Step 1 ：Given the function \(y = \tan(e^{x})\)

Step 2 ：We need to find the derivative of this function.

Step 3 ：Using the chain rule, the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 4 ：In this case, the outer function is \(\tan(x)\) and the inner function is \(e^{x}\).

Step 5 ：The derivative of \(\tan(x)\) is \(\sec^{2}(x)\) and the derivative of \(e^{x}\) is \(e^{x}\).

Step 6 ：Therefore, the derivative of the function \(y = \tan(e^{x})\) is \(y' = \sec^{2}(e^{x}) \cdot e^{x}\).

Step 7 ：However, since \(\sec^{2}(x)\) is equivalent to \(\tan^{2}(x) + 1\), we can simplify the derivative to \(y' = (\tan(e^{x})^{2} + 1) \cdot e^{x}\).

Step 8 ：Final Answer: \(y' = \boxed{(\tan(e^{x})^{2} + 1) \cdot e^{x}}\)