can knowing an improper integral dependence on first parameter help in studying dependence from second parameter?

Question

I am interested in studying the dependence on parameter $a$ of integrals of this type
$$ \int_{-\infty} ^{\infty} \frac{f(x,k)}{a^2+x^2}dx $$ whereby real $k> 0$ and $a>0$ , while about real $f(x,k)$ we only know that it is continuous and differentiable, and that the above integral exists.
Let us suppose that through other means we also know the expression for $$ \frac{d}{dk} \, \int_{-\infty} ^{\infty} \frac{f(x,k)}{a^2+x^2}dx \;\;=\;\; \int_{-\infty} ^{\infty} \frac{d f(x,k)}{dk} \, \frac{1}{a^2+x^2}dx \;\;=\;\;g(a,k) $$ Could that provide information also about $$ \frac{d}{da} \, \int_{-\infty} ^{\infty} \frac{f(x,k)}{a^2+x^2}dx \;\;=\;\;-2a \, \int_{-\infty} ^{\infty} f(x,k)\; \frac{1}{(a^2+x^2)^2}dx \;\;\;? $$ at least in terms of bounds or inequalities ?
Mentally picturing a generic $f(x,k)$ somehow alternating between pieces of positive and negative values, while trying to naively visualise how positive and negative areas are affected in $a$ following the division by $(a^2+x^2)$, the first impression is that there must be some sort of connection between the two derivatives. I imagine that this kind of problems, or similar ones, may have already been systematically studied. I would much appreciate any suggestion about any relevant literature which may be of help.