Legendre transform moves a ruler (tangent line) along a convex function, measuring how much "lag" the function accumulated while "accelerating" up to its velocity at a certain moment, relative to having constant velocity for its entire history.
The larger the function's 2nd derivative is, the smaller the transform value is.
And vice versa. Since the tranform is written in terms of the original function's derivative, not it's "x value", the derivative of the transform is inversely proportional to the derivative of the function
The larger the function's 2nd derivative is, the smaller the transform value is. And vice versa. Since the tranform is written in terms of the original function's derivative, not it's "x value", the derivative of the transform is inversely proportional to the derivative of the function
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