Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 2 - Differentiation - 2.5 Exercises: 46

Answer

$\dfrac{d^2y}{dx^2}=\dfrac{6xy-16}{x^3}$.

Work Step by Step

First Derivative: $\dfrac{d}{dx}(x^2y)-\dfrac{d}{dx}(4x)=\dfrac{d}{dx}(5)\rightarrow$ $2xy+\dfrac{dy}{dx}(x^2)-4=0\rightarrow$ $\dfrac{dy}{dx}=\dfrac{4-2xy}{x^2}$ Second Derivative: $\dfrac{d}{dx}(\dfrac{dy}{dx})=\dfrac{d}{dx}(\dfrac{4-2xy}{x^2})\rightarrow$ Using the Quotient Rule: $\dfrac{d^2y}{dx^2}=\dfrac{(-2y+\dfrac{dy}{dx}(-2x))(x^2)-(2x)(4-2xy)}{x^4}\rightarrow$ $\dfrac{2x^2y-2x^3(\dfrac{4-2xy}{x^2})-8x}{x^4}\rightarrow$ $\dfrac{6xy-16}{x^3}.$
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