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 965: 51


$\frac{\partial z}{\partial x}=f'(x)$ $\frac{\partial z}{\partial y}=g'(y)$

Work Step by Step

Given a function $z$ that is a sum of two single variable functions depending on $x$ and $y$ respectively such as: $z=f(x)+g(y)$ , the partial derivatives of $z$ can be found using the standard rules of derivatives: First, differentiate w.r.t $x$. As such, all occurrences of the variable $y$ are considered constants. $\frac{\partial z}{\partial x}=\frac{\partial}{\partial x} f(x) + \frac{\partial}{\partial x} g(y)=f'(x) + 0 = f'(x)$ The same can be done for the function w.r.t the variable $y$: $\frac{\partial z}{\partial y}=\frac{\partial}{\partial y} f(x) + \frac{\partial}{\partial y} g(y)=0 + g'(y) = g'(y)$
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