## Inductive Derivative Identities :::problem[Exponential Reciprocal Identity] Prove by induction that $$\begin{equation*} \frac{d^n}{dx^n}\left(x^{n-1}e^{1/x}\right) = (-1)^n\frac{e^{1/x}}{x^{n+1}}. \end{equation*}$$ ::: :::solution For $n=1$, direct differentiation gives the stated formula. Suppose that $$\begin{equation*} D^n\left(x^{n-1}e^{1/x}\right) = (-1)^n\frac{e^{1/x}}{x^{n+1}}. \end{equation*}$$ Differentiating once more and simplifying the product rule gives the same formula with $n$ replaced by $n+1$. Hence the identity holds for every positive integer $n$. $\square$ ::: ## Recurrence for Polynomial–Logarithmic Products :::theorem[Recurrence for an Nth Derivative] Let $P_n=D^n(x^n\log x)$ for $x>0$. Then $$\begin{equation*} P_n=nP_{n-1}+(n-1)!. \end{equation*}$$ ::: :::proof Using $x^n\log x=x\cdot x^{n-1}\log x$, apply Leibniz's formula. All derivatives of $x$ above order $1$ vanish. Comparing the remaining terms gives $P_n=nP_{n-1}+(n-1)!$. $\square$ ::: It follows that $$\begin{equation*} P_n = n!\left(\log x+1+\frac12+\frac13+\cdots+\frac1n\right). \end{equation*}$$ :::section[Method and Verification] Check the base case, state the induction hypothesis, differentiate once, and compare the result with the statement for index $n+1$. For recurrences, keep the function dependence on $n$ separate from the order of differentiation. ::: :::exercise[Practice Questions] 1. Verify the exponential-reciprocal identity for $n=1$ and $n=2$. 2. Let $P_n=D^n(x^n\log x)$. Prove $P_n=nP_{n-1}+(n-1)!$. 3. Explain why an induction proof must distinguish the function that changes with $n$ from the order of differentiation. ::: :::answer[Answers and Solution Guidance] 1. Direct differentiation confirms the stated values for $n=1$ and $n=2$. 2. Write $x^n\log x=x\cdot x^{n-1}\log x$ and apply Leibniz's formula; only two terms survive. 3. The induction index changes both the function and the derivative order, so both changes must be tracked separately. ::: :::faq[Frequently Asked Questions] Q: What is the first check in an inductive derivative identity? A: Verify the base case directly from the definition of the derivative. Q: Why are index shifts dangerous? A: A shift can change the function, the derivative order, or both, and an unstated change can invalidate the proof. Q: How should a recurrence be verified? A: Check the first two indices, derive the relation from the defining identity, and compare the resulting coefficients. ::: :::call-to-action[Complete Repeated Differentiation] subtitle: Complete the practice and review the six-lesson section. button: Review the Section|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives button-ghost: Review the Chapter|/real-analysis/functions-of-a-single-variable-ii :::