91Ó°ÊÓ

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v} .\) $$ f(x, y)=x y, \quad P(0,-2), \quad \mathrm{v}=\frac{1}{2} \mathrm{i}+\frac{\sqrt{3}}{2} \mathbf{j} $$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v} .\) $$ h(x, y)=e^{x} \sin y, P\left(1, \frac{\pi}{2}\right), \mathbf{v}=-\mathbf{i} $$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v} .\) $$ h(x, y, z)=x y z, P(2,1,1), \mathrm{v}=2 \mathbf{i}+\mathbf{j}-\mathbf{k} $$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v} .\) $$ f(x, y)=x y, P(1,1), \mathbf{u}=\left\langle\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right\rangle $$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v} .\) $$ f(x, y)=x^{2}-y^{2}, \mathbf{u}=\left\langle\frac{\sqrt{3}}{2}, \frac{1}{2}\right\rangle, \quad P(1,0) $$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v} .\) $$ f(x, y)=e^{x} \cos y, \quad \mathbf{u}=\langle 0,1\rangle, \quad P=\left(0, \frac{\pi}{2}\right) $$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v} .\) $$ f(x, y)=y^{10}, \mathbf{u}=\langle 0,-1\rangle, \quad P=(1,-1) $$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v} .\) $$ f(x, y)=\ln \left(x^{2}+y^{2}\right), \mathbf{u}=\left\langle\frac{3}{5}, \frac{4}{5}\right\rangle, \quad P(1,2) $$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v} .\) $$ f(x, y)=x^{2} y, P(-5,5), \quad \mathbf{v}=3 \mathbf{i}-4 \mathbf{j} $$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v} .\) $$ f(x, y)=y^{2}+x z, P(1,2,2), \quad \mathbf{v}=\langle 2,-1,2\rangle $$