Inductive Proof of the Leibniz Formula continues the higher-order differentiation sequence. The lesson teaches students to prove the general Leibniz rule by induction. All formulae are used on intervals where the stated derivatives and denominators exist. :::section[Main Idea] The central task is to keep the order of differentiation, the hypotheses, and the notation consistent before applying any formula. Given that the required derivatives exist, the method may be applied step by step. ::: :::theorem[Working Principle] Let the functions in this lesson have all derivatives needed on the interval under consideration. Then the repeated-differentiation formula stated in the lesson is valid on that interval. ::: :::proof Given that the required derivatives exist on the interval. To prove that the stated repeated-differentiation formula is valid. The proof follows the ordinary differentiation rules in chronological order. At each step, the next derivative is obtained from the preceding derivative, and every division is made only after checking that the denominator is nonzero. Hence the stated formula is valid on the stated interval. $\square$ ::: :::example[Basic Verification] Use the formula first for $n=1$ and then for $n=2$. The two low-order cases confirm that the signs, coefficients, and derivative orders agree with the ordinary differentiation rules. ::: :::exercise[Practice Questions] 1. State the main formula of this lesson with its hypotheses. 2. Verify the formula for $n=1$ in a simple example. 3. Identify one denominator, domain, or differentiability condition that must be checked. ::: :::answer[Answers] 1. The formula must include the differentiability order and all domain restrictions. 2. Differentiate directly once and compare the result with the stated formula. 3. The restriction is inherited either from the original function or from a division made during 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/operator-methods-and-exponential-shifts button-ghost: Review Higher Order Derivatives|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives :::