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: 19

Answer

$$z_x = \frac{1}{x + t^2}.$$ $$z_t = \frac{2t}{x + t^2}.$$

Work Step by Step

To find $z_x$, we treat $t$ as a constant and differentiate with respect to $x$. Thus $$z_x = \frac{\frac{\delta}{\delta x}(x + t^2)}{x + t^2} = \frac{1}{x + t^2}.$$ Similarly, to find $z_t$, we treat $x$ as constant and differentiate with respect to $t$. Thus $$z_t = \frac{\frac{\delta}{\delta t}(x + t^2)}{x + t^2} = \frac{2t}{x + t^2}.$$
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