91Ó°ÊÓ

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\) $$ f(x, y)=x^{2}+2 y^{2}, \theta=\frac{\pi}{6} $$

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\) $$ f(x, y)=\frac{y}{x+2 y}, \theta=-\frac{\pi}{4} $$

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\) $$ f(x, y)=\cos (3 x+y), \theta=\frac{\pi}{4} $$

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\) $$ w(x, y)=y e^{x}, \theta=\frac{\pi}{3} $$

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\) $$ f(x, y)=x \arctan (y), \quad \theta=\frac{\pi}{2} $$

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\) $$ f(x, y)=\ln (x+2 y), \quad \theta=\frac{\pi}{3} $$

For the following exercises, find the gradient. Find the gradient of \(f(x, y)=\frac{14-x^{2}-y^{2}}{3} .\) Then, find the gradient at point \(P(1,2)\)

For the following exercises, find the gradient. Find the gradient of \(f(x, y, z)=x y+y z+x z\) at point \(P(1,2,3)\)

For the following exercises, find the gradient. Find the gradient of \(f(x, y, z)\) at P and in the direction of \(\mathbf{u}\): $$ f(x, y, z)=\ln \left(x^{2}+2 y^{2}+3 z^{2}\right), P(2,1,4), \quad \mathbf{u}=\frac{-3}{13} \mathbf{i}-\frac{4}{13} \mathbf{j}-\frac{12}{13} \mathbf{k} $$

For the following exercises, find the gradient. $$ \begin{array}{l}{f(x, y, z)=4 x^{5} y^{2} z^{3}, P(2,-1,1), \quad \mathbf{u}=\frac{1}{3} \mathbf{i}+\frac{2}{3} \mathbf{j}-\frac{2}{3} \mathbf{k}}\end{array} $$