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1. Prove by induction that if y = , then d"y = (-1)"n!x¯n-1, dx" where n! = 1 · 2..... n is the product of the the first n natural numbers.

1. Prove by induction that if y = , then d"y = (-1)"n!x¯n-1, dx" where n! = 1 · 2..... n is the product of the the first n natural numbers.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem 1: Mathematical Induction Proof** Prove by induction that if \( y = \frac{1}{x} \), then \[ \frac{d^n y}{dx^n} = (-1)^n n! x^{-n-1}, \] where \( n! = 1 \cdot 2 \cdot \cdots \cdot n \) is the product of the first \( n \) natural numbers.
Transcribed Image Text:**Problem 1: Mathematical Induction Proof** Prove by induction that if \( y = \frac{1}{x} \), then \[ \frac{d^n y}{dx^n} = (-1)^n n! x^{-n-1}, \] where \( n! = 1 \cdot 2 \cdot \cdots \cdot n \) is the product of the first \( n \) natural numbers.
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