We are tasked with finding the derivative of the function:

\[
f(x) = \frac{4\tan(x)}{x}
\]

### Step-by-step Solution:

1. **Apply the Quotient Rule**:
   The function is a quotient, so we'll use the quotient rule for derivatives. The quotient rule is:

   \[
   \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}
   \]
   where \( u(x) = 4\tan(x) \) and \( v(x) = x \).

2. **Differentiate \( u(x) \)**:
   \[
   u(x) = 4\tan(x), \quad u'(x) = 4\sec^2(x)
   \]

3. **Differentiate \( v(x) \)**:
   \[
   v(x) = x, \quad v'(x) = 1
   \]

4. **Substitute into the quotient rule**:
   Now substitute \( u(x), u'(x), v(x), v'(x) \) into the quotient rule:

   \[
   f'(x) = \frac{x \cdot 4\sec^2(x) - 4\tan(x) \cdot 1}{x^2}
   \]

5. **Simplify**:
   \[
   f'(x) = \frac{4x\sec^2(x) - 4\tan(x)}{x^2}
   \]

   We can factor out the 4 in the numerator:

   \[
   f'(x) = \frac{4(x\sec^2(x) - \tan(x))}{x^2}
   \]

Thus, the derivative of \( f(x) = \frac{4\tan(x)}{x} \) is:

\[
f'(x) = \frac{4(x\sec^2(x) - \tan(x))}{x^2}
\]

Would you like further details on any part of this explanation?

### Follow-up Questions:
1. How would the derivative change if the numerator had a different trigonometric function, like \(\sin(x)\) instead of \(\tan(x)\)?
2. What is the derivative of \(\sec(x)\) and how does it apply to other quotient-rule problems?
3. Can you apply the product rule to this function instead of the quotient rule?
4. How does this derivative behave at \(x = \pi/4\) or other specific values of \(x\)?
5. How would this problem change if \(x\) were in degrees rather than radians?

#### Tip:
When using the quotient rule, it's important to simplify your derivatives as much as possible to avoid errors later in calculations!

Explore the step-by-step process to find the derivative of the function 4tan(x)/x using the quotient rule. This problem focuses on trigonometric functions and their derivatives, making it suitable for advanced high school or college-level calculus students.

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