Taking the ratio of a derivative of one function over another ...

[j_sun]

Why would a mathematician take a derivative of one function, put it over the derivative of another function, and take the ratio of the two?  Thanks.

[Melanie]

That is the technique used to determine the limit of a rational function when the function is indeterminate at the limit. Query L'Hopital's Rule.

[coco]

convergence of derivatives: comparison test, ratio test
series expansion, like Taylor series. for f'(x)/g'(x)
numerical computations of functions, f(x), f'(x), ...f^n(x) represented as partial sums