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.

Learn to differentiate the function f(x) = e^x/tan(x) by applying the quotient rule. This solution includes step-by-step differentiation of exponential and trigonometric functions, suitable for advanced calculus students in Grades 11-12.

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