# What is the graphic intuition behind Marginal rate of subsitution

So I think I understand the math, but I don't understand the intuition on a 3d graph. Why does the ratio of partial derivatives give us a tangent line on a level curve(indifference curve)? My understanding is $$\frac{\partial U(x_1,x_2)}{\partial x_1}$$ Will give a tangent line parallel to the $$U(x_1,x_2), x_1$$ plane and $$\frac{\partial U(x_1,x_2)}{\partial x_2}$$ Will give a tangent line parallel to the $$U(x_1,x_2), x_2$$ plane. So why does dividing them end up with a tangent line parallel to the $$(x_1, x_2)$$ plane?

• Try looking up "implicit differentiation". This video might also help (I love this channel.) youtube.com/watch?v=qb40J4N1fa4 – Art Aug 6 '19 at 4:33