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

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

For \(f(x,y)=e^{xy}\), find \(f_x\).

For \(f(x,y)=\sin(xy)\), find \(f_y\).

Find \(f_{xx}\) for \(f(x,y)=x^4y+3xy^2\).

Find \(f_{xy}\) for \(f(x,y)=x^2y^3+5x\).

Evaluate \(f_x(1,2)\) for \(f(x,y)=x^2y+\ln y\), where \(y>0\).

For \(f(x,y)=x^2+xy+y^2\), find the linear approximation to \(f\) at \((1,1)\).

Show that \(f_{xy}=f_{yx}\) for \(f(x,y)=x^3y^2\).

For \(z=x^2y+y^3\), estimate the change in \(z\) near \((2,1)\) when \(dx=0.1\) and \(dy=-0.2\).