Anyone interpretation of the ratio between the derivative and the square of the function?

Question

so I was wondering if anyone has ever heard of any usage, or came across at some point, of interpretation of what does $\frac{f'}{f^2}$ represents for the scalar function? Or equivalently for $\frac{f''}{f'^2}$. Any ideas are most welcome

Answer 1

If a curve is the graph of $y = f(x)$, where $f(x)$ is differentiable, then at any point $(x,f(x))$ on the curve, the slope of the line tangent to the curve is $f'(x)$ and the slope of the line perpendicular to the curve at that point is $-1/f'(x)$. $$ \frac{d}{dx} \left(-\frac{1}{f'(x)} \right) = \frac{f''(x)}{(f'(x))^2}, $$ so $f''/(f')^2$ is the rate of change (with respect to $x$) of the slope of the line perpendicular to the curve.

So in a sense it's a measure of curvature, but not a particularly satisfactory one (at least, no more satisfactory than regarding $f''$, which is the rate of change of the slope of the tangent line, as a measure of curvature).

It's also in some sense complementary to $f''$, since it relates to the perpendicular line similarly to the way in which $f''$ relates to the tangent line.

If instead we have a graph of $y = g(x)$ where $f = g'$, then $f'/f^2$ is the same as $g''/(g')^2$.