Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 3: Derivatives - Section 3.3 Differentiation Rules - Exercises 3.3: 15

Answer

(a) applying the Product Rule: $y'=3x^2+10x+2-\frac{1}{x^2}$ (b) multiplying the factors to produce a sum of simpler terms to differentiate: $y'=3x^2+10x+2-\frac{1}{x^2}$

Work Step by Step

$y=(x^2+1)(x+5+\frac{1}{x})$ (a) applying the Product Rule: $y'=f'(x)⋅g(x)+f(x)⋅g'(x)$ $y'=((2)x^{2-1}+0)(x+5+\frac{1}{x})+(x^2+1)((1)x^{1-1}+0+(-1)x^{-1-1})$ $y'=(2x)(x+5+\frac{1}{x})+(x^2+1)(1-\frac{1}{x^2})$ $y'=2x^2+10x+\frac{2x}{x}+x^2-\frac{x^2}{x^2}+1-\frac{1}{x^2}$ $y'=3x^2+10x+2-\frac{1}{x^2}$ (b) multiplying the factors to produce a sum of simpler terms to differentiate: $y=(x^2+1)(x+5+\frac{1}{x})$ $y=x^3+5x^2+\frac{x^2}{x}+x+5+\frac{1}{x}$ $y=x^3+5x^2+2x+5+\frac{1}{x}$ Derivating the function using the Power Rule: $y'=(3)x^{3-1}+(2)(5)x^{2-1}+(1)(2)x^{1-1}+(0)+(-1)x^{-1-1}$ $y'=3x^2+10x+2-x^{-2}$ $y'=3x^2+10x+2-\frac{1}{x^2}$
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