Introduction: If the value of a variable changes, then usually the function value changes as well. For a function of two variables it is however possible that there are combinations of changes of both variables that do not result in a change of the function value.
Property slope tangent line to level curve: For a change of x by Δx and a change of y by Δy=−z′x(x,y)z′y(x,y)⋅Δx, the change of x multiplied by the slope of the tangent line to the level curve at (x,y), the function value stays approximately the same. Or,
z(x+Δx,y−z′x(x,y)z′y(x,y)⋅Δx)≈z(x,y).
Property slope tangent line to level curve: For a change of x by Δx and a change of y by Δy=−z′x(x,y)z′y(x,y)⋅Δx, the change of x multiplied by the slope of the tangent line to the level curve at (x,y), the function value stays approximately the same. Or,
z(x+Δx,y−z′x(x,y)z′y(x,y)⋅Δx)≈z(x,y).