Leibniz Formula for Successive Derivatives continues the higher-order differentiation sequence. All formulae are used on intervals where the displayed expressions are defined. Repeated differentiation of a product produces a binomial pattern. :::theorem[General Leibniz Rule] Let $u$ and $v$ be $n$ times differentiable. Then $$\begin{equation*} D^n(uv)=\sum_{r=0}^{n}\binom{n}{r}u^{(n-r)}v^{(r)}. \end{equation*}$$ ::: :::example[Polynomial Factor] If $y=x^2\log x$, then derivatives of $x^2$ vanish after order $2$. Hence the Leibniz sum has only three nonzero polynomial-factor terms. ::: :::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/advanced-problems-in-repeated-differentiation button-ghost: Review Higher Order Derivatives|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives :::