To find the \( n \)-th derivative of the function \( f(x) = \frac{ax + b}{cx + d} \), we will first explore a general method using the quotient rule and some properties of derivatives.

### Step-by-Step Derivation:

#### 1. First Derivative:
Let \( f(x) = \frac{ax + b}{cx + d} \).

Using the quotient rule:
\[
f'(x) = \frac{(cx + d)(a) - (ax + b)(c)}{(cx + d)^2}
\]
Simplifying the numerator:
\[
f'(x) = \frac{a(cx + d) - c(ax + b)}{(cx + d)^2}
\]
\[
= \frac{acx + ad - acx - bc}{(cx + d)^2}
\]
\[
= \frac{ad - bc}{(cx + d)^2}
\]
Thus, the first derivative is:
\[
f'(x) = \frac{ad - bc}{(cx + d)^2}
\]

#### 2. Second Derivative:
For the second derivative, we differentiate \( f'(x) \):
\[
f''(x) = \frac{d}{dx} \left( \frac{ad - bc}{(cx + d)^2} \right)
\]
Since \( ad - bc \) is constant, we focus on differentiating \( (cx + d)^{-2} \):
\[
f''(x) = (ad - bc) \cdot \frac{d}{dx} \left( (cx + d)^{-2} \right)
\]
\[
f''(x) = (ad - bc) \cdot (-2)(cx + d)^{-3} \cdot c
\]
\[
f''(x) = -2c(ad - bc)(cx + d)^{-3}
\]

#### 3. General \( n \)-th Derivative:
From the pattern observed above, the \( n \)-th derivative involves successive powers of \( (cx + d) \). The general formula for the \( n \)-th derivative can be written as:

\[
f^{(n)}(x) = (-1)^{n-1} \cdot (n-1)! \cdot c^{n-1} \cdot (ad - bc) \cdot (cx + d)^{-(n+1)}
\]

This gives us a recursive structure where each derivative involves higher powers of \( (cx + d) \) and factorial terms.

### Final Formula:
\[
f^{(n)}(x) = (-1)^{n-1} \cdot (n-1)! \cdot c^{n-1} \cdot (ad - bc) \cdot (cx + d)^{-(n+1)}
\]

Let me know if you'd like further details or explanations on this! 

Here are five related questions to explore:
1. How would the derivative change if the function were \( f(x) = \frac{1}{cx + d} \)?
2. Can you find the first three derivatives explicitly for a specific example, like \( f(x) = \frac{3x + 2}{4x + 5} \)?
3. What happens to the \( n \)-th derivative as \( x \to \infty \)?
4. How would the formula change if \( a = c \) and \( b = d \)?
5. How do you compute the inverse function of \( f(x) = \frac{ax + b}{cx + d} \) and its derivative?

**Tip:** The quotient rule is powerful for rational functions, but identifying patterns for successive derivatives can significantly simplify calculations.

Explore the detailed step-by-step solution for finding the nth derivative of the rational function f(x) = (ax + b)/(cx + d) using the quotient rule. This solution covers key calculus concepts like derivatives, the quotient rule, and recursive patterns for higher-order derivatives. Suitable for undergraduate calculus students.

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