On partial derivatives of some summatory functions

Let $f$ be a real arithmetic function and let $g:[1,\mathrm{\infty }[\to \mathbb{R}$ be a smooth function. We describe two emblematic instances in which saddle-point estimates may be used to evaluate the frequency, on the set of integers $n⩽x$, of the event $\{f(n)⩽g(n)\}$ from those relevant to the event $\{f(n)⩽y\}$. The first example revisits Dickman’s historical contribution to the theory of friable integers. The second is concerned with the distribution of the squarefree kernel of an integer.