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: 12

Answer

First Derivative: $r'=\frac{-12}{θ^{2}}+\frac{12}{θ^{4}}-\frac{4}{θ^{5}}$ Second Derivative: $r''=\frac{24}{θ^{3}}-\frac{48}{θ^{5}}+\frac{20}{θ^{6}}$

Work Step by Step

Using the Power Rule: First Derivative: $r=\frac{12}{θ}-\frac{4}{θ^{3}}+\frac{1}{θ^{4}}$ $r'=(-1)(12)θ^{-1-1}-(-3)(4)θ^{-3-1}+(-4)θ^{-4-1}$ $r'=-12θ^{-2}+12θ^{-4}-4θ^{-5}$ $r'=\frac{-12}{θ^{2}}+\frac{12}{θ^{4}}-\frac{4}{θ^{5}}$ Second Derivative: $r'=\frac{-12}{θ^{2}}+\frac{12}{θ^{4}}-\frac{4}{θ^{5}}$ $r''=(-2)(-12)θ^{-2-1}+(-4)(12)θ^{-4-1}-(-5)(4)θ^{-5-1}$ $r''=24θ^{-3}-48θ^{-5}+20θ^{-6}$ $r''=\frac{24}{θ^{3}}-\frac{48}{θ^{5}}+\frac{20}{θ^{6}}$
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