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Problem 14. (a) Use Leibniz's product rule d (rv) = r dv+v dr to show that if n is a positive integer then d (r") = nx"- dr

Problem 14. (a) Use Leibniz's product rule d (rv) = r dv+v dr to show that if n is a positive integer then d (r") = nx"- dr

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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#14 part a please
Both Newton and Leibniz were satisfied that their calculus provided answers that agreed with what was known at the time. For exampled (r) = d (rr) = I dr+rdx 3a2 dr, results that were essentially derived by others in different ways. = 2x dæ and d () = d (x²x) = r² dr+xd (r²) = 2² +x (2x dr) = %3D Problem 14. (a) Use Leibniz's product rule d (ru) = rdv+v dr to show that if n is a positive integer then d (r") = nx"-1 de (b) Use Leibniz's product rule to derive the quotient rule y dv- v – v dy YY (c) Use the quotient rule to show that if n is a positive integer, then - (u-x) p Problem 15. Let p and q be integers with q 4 0. Show d (r) = r dr 0 = -nr n-1 dz. Leibniz also provided applications of his calculus to prove its worth. As an example he derived Snell's Law of Refraction from his calculus rules as follows. Given that light travels through air at a speed of v, and travels through water at a speed of v, the problem is to find the fastest path from point A to point B.
Transcribed Image Text:Both Newton and Leibniz were satisfied that their calculus provided answers that agreed with what was known at the time. For exampled (r) = d (rr) = I dr+rdx 3a2 dr, results that were essentially derived by others in different ways. = 2x dæ and d () = d (x²x) = r² dr+xd (r²) = 2² +x (2x dr) = %3D Problem 14. (a) Use Leibniz's product rule d (ru) = rdv+v dr to show that if n is a positive integer then d (r") = nx"-1 de (b) Use Leibniz's product rule to derive the quotient rule y dv- v – v dy YY (c) Use the quotient rule to show that if n is a positive integer, then - (u-x) p Problem 15. Let p and q be integers with q 4 0. Show d (r) = r dr 0 = -nr n-1 dz. Leibniz also provided applications of his calculus to prove its worth. As an example he derived Snell's Law of Refraction from his calculus rules as follows. Given that light travels through air at a speed of v, and travels through water at a speed of v, the problem is to find the fastest path from point A to point B.
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