Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 14 - Section 14.3 - Partial Derivatives - 14.3 Exercise: 64

Answer

$f_{yxy}=-50cos(2x+5y)$

Work Step by Step

Consider the function $f(x,y)=sin(2x+5y)$ Let us start by finding $f_{y}(x,y)$ by differentiating $f(x,y) $with respect to $y$ keeping $x$ constant. As we know $f_{y}=\frac{∂}{∂y}f(x,y) $ $=\frac{∂}{∂y}[sin(2x+5y)]$ $=5cos(2x+5y)$ Now, let us start by finding $f_{y}(x,y)$ by differentiating $f(x,y) $with respect to $x$ keeping $y$ constant. $f_{yx}=\frac{∂}{∂x}[5cos(2x+5y)]=-10sin(2x+5y)$ $f_{yxy}=\frac{∂}{∂y}[f_{yx}]$ $=\frac{∂}{∂y}[-10sin(2x+5y)]$ Hence, $f_{yxy}=-50cos(2x+5y)$
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