Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 2 - The Derivative - 2.2 The Derivative Function - Exercises Set 2.2: 18

Answer

$\frac {dy}{dx} = 4x^3 $

Work Step by Step

$ y = x^4 $ $\frac {dy}{dx} = \lim\limits_{∆x \to 0} \frac {f(x+∆x)-f(x)}{∆x} $ $\frac {dy}{dx} = \lim\limits_{∆x \to 0} \frac {(x+∆x)^4 -x^4}{∆x} $ $\frac {dy}{dx} = \lim\limits_{∆x \to 0} \frac {x^4 + 4x^3∆x+6x^2∆x^2+4x∆x^3+∆x^4-x^4}{∆x} $ $\frac {dy}{dx} = \lim\limits_{∆x \to 0} \frac {∆x(4x^3+6x^2∆x+4x∆x^2+∆x^3)}{∆x} $ $\frac {dy}{dx} = \lim\limits_{∆x \to 0} (4x^3+6x^2∆x+4x∆x^2+∆x^3) $ $\frac{dy}{dx} = 4x^3+6x^2(0)+4x^3(0)^2+(0)^3 $ $\frac{dy}{dx} = 4x^3 $
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