Find \(\nabla f\) for \(f(x,y)=x^2+3y^2\).

Evaluate \(\nabla f(1,2)\) for \(f(x,y)=x^2y+y\).

Find the directional derivative of \(f(x,y)=x^2+y^2\) at \((1,2)\) in the unit direction \(\mathbf{u}=(3/5,4/5)\).

For \(f(x,y)=xy\), find the direction of greatest increase at \((2,3)\).

Find the maximum rate of change of \(f(x,y)=x^2-y^2\) at \((3,4)\).

For \(f(x,y)=x^2+y^2\), find a normal vector to the level curve through \((1,1)\).

Find a tangent line to the level curve \(x^2+y^2=5\) at \((1,2)\).

For \(f(x,y,z)=x^2+yz\), find \(\nabla f\).

Find the directional derivative of \(f(x,y)=e^x\cos y\) at \((0,0)\) in the direction of \(\mathbf{v}=(1,1)\).

At a point \((a,b)\), \(\nabla f(a,b)=(0,0)\). What is the directional derivative of \(f\) at \((a,b)\) in any unit direction \(\mathbf{u}\)?