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To find the derivative of the function \( f(x) = \frac{\tan(x)}{4 + 5x} \), we need to apply the quotient rule. The quotient rule for derivatives states:

\[
\left( \frac{u}{v} \right)' = \frac{v u' - u v'}{v^2}
\]

where:
- \( u = \tan(x) \)
- \( v = 4 + 5x \)

Let's find \( u' \) and \( v' \):

1. **Derivative of \( u = \tan(x) \):**
   \[
   u' = \sec^2(x)
   \]

2. **Derivative of \( v = 4 + 5x \):**
   \[
   v' = 5
   \]

Now, applying the quotient rule:

\[
f'(x) = \frac{(4 + 5x) \sec^2(x) - \tan(x) \cdot 5}{(4 + 5x)^2}
\]

So the derivative \( f'(x) \) is:

\[
f'(x) = \frac{(4 + 5x) \sec^2(x) - 5 \tan(x)}{(4 + 5x)^2}
\]

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. How would the derivative change if the function was \( f(x) = \frac{\sin(x)}{4 + 5x} \)?
2. What is the value of \( f'(x) \) at \( x = 0 \)?
3. How do we interpret the behavior of \( f(x) \) based on its derivative?
4. Can we simplify the expression for \( f'(x) \) further?
5. How would you apply the product rule if the function were instead \( f(x) = \tan(x) \cdot (4 + 5x) \)?

**Tip:** Memorizing the basic trigonometric derivatives, like \( \sec^2(x) \) for \( \tan(x) \), is useful for calculus problems involving trigonometric functions.

Find the derivative of the function f(x) = tan(x) / (4 + 5x).

Learn how to find the derivative of the function f(x) = tan(x) / (4 + 5x) using the quotient rule. This problem demonstrates calculus concepts, including trigonometric derivatives and the application of the quotient rule, suitable for high school students in Grades 11-12.

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