91Ó°ÊÓ

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(Q .\) $$ f(x, y, z)=\frac{y}{x+z}, P(2,1,-1), \quad Q(-1,2,0) $$

For the following exercises, find the derivative of the function at \(P\) in the direction of \(\mathbf{u} .\) $$ f(x, y)=-7 x+2 y, P(2,-4), \quad \mathbf{u}=4 \mathbf{i}-3 \mathbf{j} $$

For the following exercises, find the derivative of the function at \(P\) in the direction of \(\mathbf{u} .\) $$ f(x, y)=\ln (5 x+4 y), P(3,9), \quad \mathbf{u}=6 \mathbf{i}+8 \mathbf{j} $$

[T] Use technology to sketch the level curve of \(f(x, y)=4 x-2 y+3\) that passes through \(P(1,2)\) and draw the gradient vector at \(P .\)

[T] Use technology to sketch the level curve of \(f(x, y)=x^{2}+4 y^{2}\) that passes through \(P(-2,0)\) and draw the gradient vector at \(P .\)

For the following exercises, find the gradient vector at the indicated point. $$ f(x, y)=x y^{2}-y x^{2}, P(-1,1) $$

For the following exercises, find the gradient vector at the indicated point. $$ f(x, y)=x e^{y}-\ln (x), P(-3,0) $$

For the following exercises, find the gradient vector at the indicated point. $$ f(x, y, z)=x y-\ln (z), P(2,-2,2) $$

For the following exercises, find the gradient vector at the indicated point. $$ f(x, y, z)=x \sqrt{y^{2}+z^{2}}, P(-2,-1,-1) $$

For the following exercises, find the derivative of the function. \(f(x, y)=e^{x y}\) at point \((6,7)\) in the direction the function increases most rapidly