Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.4 The Product and Quotient Rules - 3.4 Exercises: 67

Answer

$f'(x)=\dfrac{2(100x^{3}+40x^{2}+4x-1)}{(5x+1)^{2}}$

Work Step by Step

$f(x)=4x^{2}-\dfrac{2x}{5x+1}$ Evaluate the derivative term by term. Use the quotient rule to evaluate the derivative of the second term: $f'(x)=(4x^{2})'-\Big(\dfrac{2x}{5x+1}\Big)'=...$ $...=8x-\dfrac{(5x+1)(2x)'-(2x)(5x+1)'}{(5x+1)^{2}}=...$ Evaluate the derivatives indicated and simplify: $...=8x-\dfrac{2(5x+1)-5(2x)}{(5x+1)^{2}}=8x-\dfrac{10x+2-10x}{(5x+1)^{2}}=...$ $...=8x-\dfrac{2}{(5x+1)^{2}}=\dfrac{8x(5x+1)^{2}-2}{(5x+1)^{2}}=...$ $...=\dfrac{8x(25x^{2}+10x+1)-2}{(5x+1)^{2}}=\dfrac{200x^{3}+80x^{2}+8x-2}{(5x+1)^{2}}=...$ $...=\dfrac{2(100x^{3}+40x^{2}+4x-1)}{(5x+1)^{2}}$
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