>>11275033Think of the function x maps to (g(x), f(x)), a function of R into R^2. This is a curve. What does f'g - fg' measure? It measures the turning rate about the origin, it is a sort of curl (why? if in the first quadrant, we move cw if f' < 0 and g' > 0, then f'g - g'f < 0, and ccw if f' > 0 and g' < 0, then f'g - g'f > 0. check for other quadrants to see that the sign always matches the direction of rotation, + for ccw, - for cw). Now, we have a measure for the rate of rotation but need to normalize it to get the rate of change of the slope f/g. What is our normalization factor? Think dimensional analysis. If f is in units of length and g units of mass, then f'g and g'f both have length*mass/time units, while what we want, (f/g)', has length/mass/time units. So the only way we can normalize using what we've got is by some constant multiple of 1/g^2, which has units 1/mass^2. Now, trying on a simple example, x/x, we see that the constant is 1.
The normalization business is not easy to intuit but the top expression is just how you look at curl of a 2d vector field, so you should get comfortable with why this is related to division. Always think slope.
