# Chapter 15 - Differentiation in Several Variables - 15.5 The Gradient and Directional Derivatives - Exercises - Page 802: 47

$$0.5217 x+0.7826 y-1.2375 z=-5.309$$

#### Work Step by Step

Given $$\ln \left[1+4 x^{2}+9 y^{4}\right]-0.1 z^{2}=0, \quad P=(3,1,6.1876)$$ Consider $f(x,y,z)=\ln \left[1+4 x^{2}+9 y^{4}\right]-0.1 z^{2}$, since \begin{align*} \nabla f&=\left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right\rangle\\ &=\left\langle\frac{8 x}{1+4 x^{2}+9 y^{4}}, \frac{36 y^{3}}{1+4 x^{2}+9 y^{4}},-0.2 z\right)\\ \nabla f_{P}&=\langle -5, 9, 6\sqrt{e}\rangle \end{align*} Then the equation of the tangent plane at $P$ is given by \begin{align*} \nabla f_{P} \cdot\langle x-x_1, y-y_1, z-z_1\rangle&= 0\\ \langle 0.5217,0.7826,-1.2375\rangle\cdot\langle x-3, y-1, z-6.1876\rangle&=0\\ 0.5217(x-3)+0.7826(y-1)-1.2375(z-6.1876)&=0\\ 0.5217 x+0.7826 y-1.2375 z&=-5.309 \end{align*}

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