Introduction

Understanding the nth derivative of a function is crucial in various fields of mathematics, physics, and engineering. This article explores how to find the nth derivative of a specific function through the method of partial fraction decomposition. The process involves a series of algebraic and analytical steps, shedding light on the underlying principles that make this method so powerful.

Step-by-Step Guide to Finding the nth Derivative

This guide will walk through the detailed steps involved in finding the nth derivative of the function $f(x) frac{4x}{(x-1)^2(x 1)}$ through partial fraction decomposition and differentiation.

Step 1: Partial Fraction Decomposition

The first step is to decompose the function into a sum of simpler fractions. We start by expressing the function as:

$f(x) frac{A}{x-1} frac{B}{(x-1)^2} frac{C}{x 1}$

Multiplying through by $(x-1)^2(x 1)$, we get:

$4x A(x-1)(x 1) B(x 1) C(x-1)^2$

Step 2: Setting Up the System of Equations

Expanding and combining like terms, we form the following system of equations by equating coefficients:

$A - C 0$ $B - 2C 4$ $B - C 0$

Step 3: Solving the System

Solving these equations step-by-step:

$A C$ $B - 2C 4$ becomes $B 4 2C$ $4 2C - C 0$ implies $C -4$, $A -4$, and $B 4$

Step 4: Writing the Partial Fraction Decomposition

With the values of $A$, $B$, and $C$, we can write:

$f(x) frac{-4}{x-1} frac{4}{(x-1)^2} - frac{4}{x 1}$

Step 5: Taking the nth Derivative

Now, we need to find the nth derivative of each term:

$frac{d^n}{dx^n} left( frac{-4}{x-1} right) frac{-4 cdot n!}{(x-1)^{n-1}}$ $frac{d^n}{dx^n} left( frac{4}{(x-1)^2} right) frac{4 cdot n! cdot (n-1)!}{(x-1)^{n-2}}$ $frac{d^n}{dx^n} left( -frac{4}{x 1} right) -frac{4 cdot n!}{(x 1)^{n-1}}$

Step 6: Combining the Results

Combining these results, we get:

$f^{(n)}(x) frac{-4 cdot n!}{(x-1)^{n-1}} frac{4 cdot n! cdot (n-1)!}{(x-1)^{n-2}} - frac{4 cdot n!}{(x 1)^{n-1}}$

Conclusion

The nth derivative of the function $f(x) frac{4x}{(x-1)^2(x 1)}$ can be expressed as:

$f^{(n)}(x) frac{4n!}{3(x-1)^{n-1}} frac{4n!(n-1)}{3(x-1)^{n-2}} - frac{4n!}{3(x 1)^{n-1}}$

This method not only simplifies complex derivatives but also provides a structured approach for dealing with such problems. By mastering the technique of partial fraction decomposition and differentiation, one can enhance their analytical skills and tackle a wide range of mathematical problems.