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: 63


$f_{xyx}=24x^{2}y-6x$ $f_{xxx}=24xy^{2}-6y$ and $f_{xyx}=24x^{2}y-6x$

Work Step by Step

Consider the function $f(x,y)=x^{4}y^{2}-x^{3}y$ Let us start by finding $f_{x}(x,y)$ by differentiating $f(x,y) $with respect to $x$ keeping $y$ constant. As we know $f_{x}=\frac{∂}{∂x}f(x,y) $ $=\frac{∂}{∂x}[x^{4}y^{2}-x^{3}y]$ $=4x^{3}y^{2}-3x^{2}y$ $f_{xx}=\frac{∂}{∂x}[4x^{3}y^{2}-3x^{2}y]=12x^{2}y^{2}-6xy$ Thus, $f_{xxx}=\frac{∂}{∂x}[12x^{2}y^{2}-6xy]=24xy^{2}-6y$ $f_{xy}=\frac{∂}{∂y}[4x^{3}y^{2}-3x^{2}y]=8x^{3}y-3x^{2}$ $f_{xyx}=\frac{∂}{∂x}[8x^{3}y-3x^{2}]$ Hence, $f_{xyx}=24x^{2}y-6x$ $f_{xxx}=24xy^{2}-6y$ and $f_{xyx}=24x^{2}y-6x$
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