We need to find the \(n\)-th derivative of the function \( f(x) = \frac{1}{1 - 5x + 6x^2} \).

### Step 1: Simplify the expression
The denominator can be factored as:
\[
1 - 5x + 6x^2 = (1 - 2x)(1 - 3x)
\]
So, the function becomes:
\[
f(x) = \frac{1}{(1 - 2x)(1 - 3x)}
\]

### Step 2: Apply partial fraction decomposition
We express \( f(x) \) as a sum of partial fractions:
\[
\frac{1}{(1 - 2x)(1 - 3x)} = \frac{A}{1 - 2x} + \frac{B}{1 - 3x}
\]
Multiplying both sides by \( (1 - 2x)(1 - 3x) \) and solving for \( A \) and \( B \):
\[
1 = A(1 - 3x) + B(1 - 2x)
\]
Expanding the right-hand side:
\[
1 = A - 3Ax + B - 2Bx
\]
Group terms by powers of \(x\):
\[
1 = (A + B) + (-3A - 2B)x
\]
This gives two equations:
1. \( A + B = 1 \)
2. \( -3A - 2B = 0 \)

Solving these, we get \( A = 2 \) and \( B = -1 \). So the partial fraction decomposition is:
\[
f(x) = \frac{2}{1 - 2x} - \frac{1}{1 - 3x}
\]

### Step 3: Find the \(n\)-th derivative of each term
The general formula for the \(n\)-th derivative of \( \frac{1}{1 - ax} \) is:
\[
\frac{d^n}{dx^n} \left( \frac{1}{1 - ax} \right) = n! a^n
\]

Thus, for \( f(x) = \frac{2}{1 - 2x} - \frac{1}{1 - 3x} \):
1. The \(n\)-th derivative of \( \frac{2}{1 - 2x} \) is:
   \[
   2 \cdot n! \cdot 2^n
   \]
2. The \(n\)-th derivative of \( \frac{1}{1 - 3x} \) is:
   \[
   -n! \cdot 3^n
   \]

### Step 4: Combine the results
The \(n\)-th derivative of \( f(x) \) is:
\[
f^{(n)}(x) = 2 \cdot n! \cdot 2^n - n! \cdot 3^n = n! \left( 2^{n+1} - 3^n \right)
\]

Thus, the \(n\)-th derivative of \( f(x) \) is:
\[
f^{(n)}(x) = n! \left( 2^{n+1} - 3^n \right)
\]

### Do you want details or have any questions?

Here are 5 related questions:
1. How do you apply partial fraction decomposition in other cases?
2. What are the general rules for the \(n\)-th derivative of more complex functions?
3. How does Taylor expansion relate to derivatives?
4. How can generating functions help in solving recurrence relations in derivatives?
5. How can Laplace transforms be used in derivative problems?

**Tip:** When finding the \(n\)-th derivative of a function, use known formulas for common patterns like rational functions or exponentials to save time.

Learn how to find the nth derivative of the function 1 / (1 - 5x + 6x²) using partial fraction decomposition and calculus. This step-by-step guide covers key concepts in calculus and provides the formula for higher-order derivatives. Suitable for undergraduate students.

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