Rational Decomposition for Successive Differentiation continues the higher-order differentiation sequence. All formulae are used on intervals where the displayed expressions are defined. A rational function is a quotient $P(x)/Q(x)$ with $Q(x)\ne0$. It is proper when $\deg P<\deg Q$. :::definition[Partial-Fraction Decomposition] A partial-fraction decomposition writes a proper rational function as a sum of fractions whose denominators are powers of irreducible factors of $Q(x)$. ::: :::example[Improper Rational Function] $$\begin{equation*} \frac{x^3+1}{x^2-1}=x+\frac{x+1}{x^2-1}. \end{equation*}$$ The polynomial part is differentiated directly, while the proper part is decomposed before repeated differentiation. ::: :::exercise[Practice Questions] 1. State the main result of this lesson with its hypotheses. 2. Apply the method to a simple example. 3. Identify one restriction that must be checked. ::: :::answer[Answers] 1. The result must include the differentiability and nonzero-denominator conditions. 2. Work term by term and verify a low-order case. 3. The restriction comes from the original domain or from a division made in the derivation. ::: :::faq[Frequently Asked Questions] Q: Why are hypotheses stated before the formula? A: They identify the interval, differentiability order, and excluded values needed for a valid calculation. Q: What is the common mistake? A: The common mistake is to use the displayed formula after losing a domain or nonzero-denominator condition. Q: How can I check my result? A: Substitute a small value of $n$ or differentiate once directly and compare with the formula. ::: :::call-to-action[Continue Learning] subtitle: Review the result, complete the practice questions, and continue through the higher-order derivatives sequence. button: Next Lesson|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives/linear-factor-partial-fractions button-ghost: Review Higher Order Derivatives|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives :::