Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.3 Partial Derivatives - 14.3 Exercises - Page 967: 104



Work Step by Step

The derivative of a function at the some point can be defined as: $\lim\limits_{h \to 0}\dfrac{f(x+h,y)-f(x,y)}{h}=\dfrac{\partial f }{\partial x}$ At $(x,y)=(0,0)$ $\lim\limits_{h \to 0}\dfrac{f(0+h,0)-f(0,0)}{h}=\dfrac{\partial f }{\partial x}(0,0)$ $\lim\limits_{h \to 0}\dfrac{\sqrt [3] {h^3+0^2}-\sqrt [3] {0^3+0^2}}{h}=\dfrac{\partial f }{\partial x}(0,0)$ After solving the above equations, we have $\lim\limits_{h \to 0}\dfrac{h}{h}=\dfrac{\partial f }{\partial x}(0,0)=1$
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