To find the **n-th derivative** of the function \( f(x) = \frac{ax + b}{cx + d} \), we can begin by calculating the first few derivatives and observing any patterns.

### Step 1: Rewrite the function in a more convenient form
The function \( f(x) = \frac{ax + b}{cx + d} \) is a **rational function**. Let's express it using quotient rule, which is necessary for differentiation.

### Step 2: First derivative using quotient rule
The quotient rule is given by:
\[
\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}
\]
Here, \( u = ax + b \) and \( v = cx + d \), so:
- \( u' = a \)
- \( v' = c \)

Now apply the quotient rule:
\[
f'(x) = \frac{(a)(cx + d) - (ax + b)(c)}{(cx + d)^2}
\]
Simplifying the numerator:
\[
f'(x) = \frac{acx + ad - acx - bc}{(cx + d)^2}
\]
\[
f'(x) = \frac{ad - bc}{(cx + d)^2}
\]

### Step 3: Second derivative
Now differentiate \( f'(x) = \frac{ad - bc}{(cx + d)^2} \) again using the chain rule:
\[
f''(x) = (ad - bc) \cdot \left( \frac{-2c}{(cx + d)^3} \right)
\]
\[
f''(x) = \frac{-2c(ad - bc)}{(cx + d)^3}
\]

### Step 4: Third derivative
Now differentiate \( f''(x) \):
\[
f'''(x) = -2c(ad - bc) \cdot \left( \frac{-3c}{(cx + d)^4} \right)
\]
\[
f'''(x) = \frac{6c^2(ad - bc)}{(cx + d)^4}
\]

### Step 5: General pattern for the n-th derivative
From the first few derivatives, we can observe a pattern:

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

### Final formula:
Thus, the **n-th derivative** of \( f(x) = \frac{ax + b}{cx + d} \) is:

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

Would you like more details on any step, or do you have any questions?

### Related Questions:
1. How do you differentiate rational functions?
2. What is the quotient rule and how is it applied?
3. Can you explain the chain rule for composite functions?
4. How do factorials appear in higher-order derivatives?
5. What are some applications of higher-order derivatives?

### Tip:
For rational functions, higher-order derivatives often follow a recognizable pattern after differentiating the first few times. Look for factorials and powers in the formula.

Explore how to find the n-th derivative of the function (ax + b) / (cx + d) using the quotient and chain rules. This step-by-step solution identifies patterns in higher-order derivatives and explains the formula for n-th derivatives of rational functions. Suitable for students in undergraduate calculus.

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