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!