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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?

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    $\begingroup$ Try looking up "implicit differentiation". This video might also help (I love this channel.) youtube.com/watch?v=qb40J4N1fa4 $\endgroup$ – Art Aug 6 at 4:33
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This isn't a perfect answer to your question, but I tend to show all my students pages 1-4 of Dixit's Optimization book. It describes exactly what the MRS is doing and why it must be tangent to the budget constraint.

https://www.scribd.com/document/390573490/Dixit-Optimization-in-Economic-Theory-pdf (pages 1-3 here unless you sign in)

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