Find the derivative of the furiction.
\[
f(x)=5 \tan ^{-1}\left(e^{x}\right)
\]

Step 1 ：Given the function \(f(x)=5 \tan ^{-1}\left(e^{x}\right)\)

Step 2 ：We need to find its derivative using the chain rule and the derivative of the inverse tangent function.

Step 3 ：The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 4 ：The derivative of the inverse tangent function is \(\frac{1}{1+x^2}\).

Step 5 ：First, let's find the derivative of the outer function, which is \(5 \tan ^{-1}(x)\). The derivative of this function is \(\frac{5}{1+x^2}\).

Step 6 ：Next, let's find the derivative of the inner function, which is \(e^x\). The derivative of this function is also \(e^x\).

Step 7 ：Now, we can apply the chain rule. The derivative of the composite function is the product of the derivative of the outer function and the derivative of the inner function.

Step 8 ：So, the derivative of \(f(x)=5 \tan ^{-1}\left(e^{x}\right)\) is \(f'(x) = \frac{5}{1+(e^x)^2} \cdot e^x = \frac{5e^x}{1+e^{2x}}\).

Step 9 ：\(\boxed{f'(x) = \frac{5e^x}{1+e^{2x}}}\) is the simplest form of the derivative.