Applications 2: Marginal rate of subsitution

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 $\Delta x$ and a change of $y$ by $\Delta y =  -\dfrac{z_x'(x,y)}{z_y'(x,y)}  \cdot \Delta x $, 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+\Delta x, y  -\frac{z_x'(x,y)}{z_y'(x,y)} \cdot \Delta x) \approx z(x,y).