Calculus (3rd Edition)

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

Chapter 15 - Differentiation in Several Variables - 15.5 The Gradient and Directional Derivatives - Exercises - Page 801: 19

Answer

$$0$$

Work Step by Step

Given $$ g(x, y, z)=x y z^{-1}, \quad \mathbf{r}(t)=\left\langle e^{t}, t, t^{2}\right\rangle, \quad t=1 $$ since $\mathbf{r}(1)=\langle e,1,1\rangle$ and $$ \begin{aligned} \nabla g &=\left\langle\frac{\partial g}{\partial x}, \frac{\partial g}{\partial y},\frac{\partial g}{\partial z}\right\rangle \\ &=\langle yz^{-1}, xz^{-1},-xyz^{-2} \rangle \\ \mathbf{r}^{\prime}(t) &=\left\langle e^t, 1,2t\right\rangle \end{aligned} $$ Then \begin{aligned} \left.\frac{d}{d t} f(\mathbf{r}(t))\right|_{t=1}&=\nabla f_{\mathbf{r}(1)} \cdot \mathbf{r}^{\prime}(1)\\ &=\langle 1,e,-e \rangle \cdot\langle e,1,2 \rangle \\ &=e+e-2e=0 \end{aligned}
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