Linear-Factor Partial Fractions in Higher Derivatives continues the higher-order differentiation sequence. All formulae are used on intervals where the displayed expressions are defined. Linear factors give direct repeated-derivative terms. :::theorem[Distinct Linear Factors] If $$\begin{equation*} \frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b},\qquad a\ne b, \end{equation*}$$ then $$\begin{equation*} D^n\left(\frac{P(x)}{(x-a)(x-b)}\right)=(-1)^n n!\left(\frac{A}{(x-a)^{n+1}}+\frac{B}{(x-b)^{n+1}}\right). \end{equation*}$$ ::: :::remark[Repeated Linear Factor] For $r\in\mathbb{N}$ and $x\ne a$, $$\begin{equation*} D^n\left((x-a)^{-r}\right)=(-1)^n\frac{(r+n-1)!}{(r-1)!}(x-a)^{-r-n}. \end{equation*}$$ ::: :::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/quadratic-factor-partial-fractions button-ghost: Review Higher Order Derivatives|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives :::