proving gradient of a function is always perpendicular to the contour lines Can someone give an explanation of how such a proof would go, given a function
example:   $y = f(x)$

I have an intuitive (not formal, just to get a good image) answer for you: What is a gradient? It is the direction of fastest increase for the  function $g$. What is a contour line? It is a line on which $g$ has the same value everywhere. But what direction must the gradient have with respect to the contour line to prevent the function $g$ from changing its value? - Perpendicular.
