Calculus, 10th Edition (Anton)

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

Chapter 13 - Partial Derivatives - 13.1 Functions Of Two Or More Variables - Exercises Set 13.1 - Page 914: 2


(a) $f(t,t^2)=2t$, (b) $f(x,x^2)=2x$, (c) $f(2y^2,4y)=2y(y+1)$

Work Step by Step

(a) Substitute $x=t$ and $y=t^2:$ $$f(t,t^2)=t+\sqrt[3]{t\cdot t^2}=t+\sqrt[3]{t^3}=t+t=2t$$ (b) Substitute $x=x$ and $y=x^2:$ $$f(x,x^2)=x+\sqrt[3]{x\cdot x^2}=x+\sqrt[3]{x^3}=x+x=2x$$ (c) Substitute $x=2y^2$ and $y=4y:$ $$f(2y^2,4y)=2y^2+\sqrt[3]{2y^2\cdot4y}=2y^2+\sqrt[3]{(2y)^3}=2y^2+2y=2y(y+1)$$
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