To differentiate $f(x) = \frac{e^x}{\tan(x)}$, we will use the quotient rule, which is applied to functions of the form $\frac{u(x)}{v(x)}$, where $u(x)$ and $v(x)$ are differentiable functions. The quotient rule is given by:

$\left( \frac{u(x)}{v(x)} \right)' = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$

Here, $u(x) = e^x$ and $v(x) = \tan(x)$.

1. Differentiate $u(x) = e^x$: $u'(x) = e^x$

2. Differentiate $v(x) = \tan(x)$: $v'(x) = \sec^2(x)$

Now, apply the quotient rule:

$f'(x) = \frac{e^x \cdot \tan(x) - e^x \cdot \sec^2(x)}{\tan^2(x)}$

Factor out $e^x$ from the numerator:

$f'(x) = \frac{e^x \left( \tan(x) - \sec^2(x) \right)}{\tan^2(x)}$

So, the derivative of $f(x) = \frac{e^x}{\tan(x)}$ is:

$f'(x) = \frac{e^x \left( \tan(x) - \sec^2(x) \right)}{\tan^2(x)}$

Would you like further details on any part of this?

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Here are 5 follow-up questions to enhance your understanding:

1. How do we derive the quotient rule?
2. What is the derivative of $\sec(x)$ and how is it related to $\tan(x)$?
3. Can you apply the product rule to a different function involving $e^x$?
4. How would you differentiate $\ln(\tan(x))$?
5. What happens when we apply implicit differentiation to a function like $y = \tan(x)$?

Tip: Always check the domain of the function before differentiating, especially when it involves trigonometric functions like $\tan(x)$, which has vertical asymptotes.