Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.3 Partial Derivatives - Exercises - Page 781: 34


$$ z_x= y^x\ln y,\quad z_y= \frac{x}{y}y^x. $$

Work Step by Step

Recall the product rule: $(uv)'=u'v+uv'$ Recall that $(\ln x)'=\dfrac{1}{x}$ Since $ z=y^x $, applying $\ln $ on both sides, we get $$\ln z=x\ln y.$$ Now, by using the product rule, we have $$ \frac{z_x}{z}=\ln y \Longrightarrow z_x=z\ln y=y^x\ln y,\\ \frac{z_y}{z}=\frac{x}{y} \Longrightarrow z_y=z \frac{x}{y}=\frac{x}{y}y^x. $$
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