Abstract Algebra: Theory and Applications · Problem 2.4.8

Groups — Thomas Judson

Mathematics — Abstract Algebra · leibniz-rule-induction

Problem

Prove the Leibniz rule for f(n)(x)f^{(n)}(x), where f(n)f^{(n)} is the nnth derivative of ff; that is, show that (fg)(n)(x)=∑k=0n(nk)f(k)(x)g(n−k)(x).(fg)^{(n)}(x) = \sum_{k=0}^{n} \binom{n}{k} f^{(k)}(x) g^{(n-k)}(x). Hint. Follow the proof in Example 2.1.4.

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