## Proper Rational Functions and Distinct Linear Factors ### Definition: Partial Fractions Let $f(x)$ and $\phi(x)$ be polynomials. A proper rational function $f(x)/\phi(x)$ may be expressed as a sum of simpler rational functions called **partial fractions**. ### Formula: Linear Factors without Repetition For distinct linear factors, $$\begin{equation*} \frac{f(x)} {(a_1x+b_1)(a_2x+b_2)\cdots(a_kx+b_k)} = \frac{A_1}{a_1x+b_1} + \frac{A_2}{a_2x+b_2} +\cdots+ \frac{A_k}{a_kx+b_k}. \end{equation*}$$ ### Worked Example: Distinct Linear Factors Resolve $$\begin{equation*} \frac{x^2-6}{x^3-x^2-2x} \end{equation*}$$ into partial fractions and hence find its nth derivative. ### Solved Problem: Polynomial Division Followed by Partial Fractions Verify that, for $n\geq1$, $$\begin{equation*} D^n\left( \frac{x^4+7x^3+21x^2+33x+20} {x^3+6x^2+11x+6} \right) = (-1)^n n! \left[ \frac{1}{(x+1)^{n+1}} + \frac{2}{(x+2)^{n+1}} + \frac{3}{(x+3)^{n+1}} \right]. \end{equation*}$$ ## Repeated Linear Factors ### Formula: Decomposition for a Repeated Linear Factor For a factor $(ax+b)^k$, the corresponding partial fractions are $$\begin{equation*} \frac{A_1}{(ax+b)^k} + \frac{A_2}{(ax+b)^{k-1}} +\cdots+ \frac{A_k}{ax+b}. \end{equation*}$$ ### Worked Example: Repeated Linear Factor Resolve $$\begin{equation*} \frac{x-2}{x^3-x^2-5x-3} \end{equation*}$$ into partial fractions and hence find its nth derivative. ## Irreducible Quadratic Factors ### Formula: Non-Repeated Irreducible Quadratic Factor For an irreducible quadratic factor $ax^2+bx+c$, the corresponding partial fraction has the form $$\begin{equation*} \frac{Ax+B}{ax^2+bx+c}. \end{equation*}$$ ### Worked Example: Linear and Irreducible Quadratic Factors Resolve $$\begin{equation*} \frac{13x}{(x-3)(x^2+x+1)} \end{equation*}$$ into partial fractions and determine its nth derivative. ### Formula: Repeated Irreducible Quadratic Factor For a repeated factor $(ax^2+bx+c)^k$, include one linear numerator over every power of the quadratic factor. ### Worked Example: Repeated Irreducible Quadratic Factor Resolve $$\begin{equation*} \frac{x^4+4x^2+2x+9} {(x-1)(x^2+x+2)^2} \end{equation*}$$ into partial fractions. :::section[Method and Verification] First make the rational function proper. Resolve it into elementary fractions, differentiate each term with the linear-reciprocal formula, and combine the results without altering the excluded points of the original domain. ::: :::exercise[Practice Questions] 1. Resolve $1/[x(x+1)]$ into partial fractions and find its nth derivative. 2. Find $D^n\{1/[(x-a)(x-b)]\}$ for $a\ne b$. 3. Explain why polynomial division must precede partial fractions for an improper rational function. ::: :::answer[Answers and Solution Guidance] 1. $1/[x(x+1)]=1/x-1/(x+1)$, hence $D^n=(-1)^n n![x^{-n-1}-(x+1)^{-n-1}]$. 2. $D^n\{1/[(x-a)(x-b)]\}=\dfrac{(-1)^n n!}{a-b}\{(x-a)^{-n-1}-(x-b)^{-n-1}\}$. 3. Partial-fraction decomposition applies directly to a proper rational function. Division separates the polynomial part, whose derivatives are handled independently. ::: :::faq[Frequently Asked Questions] Q: What is the central method in partial fractions in repeated differentiation? A: The method is to identify the applicable formula or decomposition, state its hypotheses, and then carry out the derivative calculation in the required order. Q: How should a general nth-derivative formula be verified? A: Check one or two initial orders and differentiate the proposed expression once to confirm the formula with $n$ replaced by $n+1$. Q: Which restrictions must be recorded? A: Record every domain condition, excluded denominator value, differentiability requirement, and index restriction used by the formula. ::: :::call-to-action[Continue Learning] subtitle: Complete the practice, verify the main result, and continue to the next repeated-differentiation lesson. button: Next Lesson|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives/leibniz-formula-for-successive-derivatives button-ghost: Review Higher-Order Derivatives|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives :::