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3. The Leibniz rule for the nth derivative of a product is given by dn dx" n (f(x) g(x)) = f(")(x)g(x) + 7+ f("-¹)(x)g '(x) 1! n(n − 1) fƒ(n-2)(x)g(²)(x) + + 2! where the coefficients are the same as the coefficients in the binomial expansion. n(n − 1) 2! An-3 B³+ + B". n (A + B)" = A" + " A"¯¹ B + ... + n(n-1)(n-2) 3! Use mathematical induction to establish the Leibniz rule. + f(x) g(n)(x), An-2B²

3. The Leibniz rule for the nth derivative of a product is given by dn dx" n (f(x) g(x)) = f(")(x)g(x) + 7+ f("-¹)(x)g '(x) 1! n(n − 1) fƒ(n-2)(x)g(²)(x) + + 2! where the coefficients are the same as the coefficients in the binomial expansion. n(n − 1) 2! An-3 B³+ + B". n (A + B)" = A" + " A"¯¹ B + ... + n(n-1)(n-2) 3! Use mathematical induction to establish the Leibniz rule. + f(x) g(n)(x), An-2B²

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. The Leibniz rule for the nth derivative of a product is given by d" n (f(x)g(x)) = f("(x)g(x) +++ f("−¹)(x)g'(x) dx" 1! n(n − 1) ƒ‹n−2)(x)g(²)(x) + + 2! where the coefficients are the same as the coefficients in the binomial expansion. n(n-1) An-²B² 2! ... + f(x)g(n)(x), n (A + B)" = A" + — A"¯¹ B + 1! n(n − 1)(n-2) 3! Use mathematical induction to establish the Leibniz rule. A-3B³+ + B".
Transcribed Image Text:3. The Leibniz rule for the nth derivative of a product is given by d" n (f(x)g(x)) = f("(x)g(x) +++ f("−¹)(x)g'(x) dx" 1! n(n − 1) ƒ‹n−2)(x)g(²)(x) + + 2! where the coefficients are the same as the coefficients in the binomial expansion. n(n-1) An-²B² 2! ... + f(x)g(n)(x), n (A + B)" = A" + — A"¯¹ B + 1! n(n − 1)(n-2) 3! Use mathematical induction to establish the Leibniz rule. A-3B³+ + B".
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