I have a set of points $(x_i,y_i)\in{\mathbb R}_+\times{\mathbb R}$, $i=1,...,n$, ($x_i$ are the independent variables and $y_i$ are the dependent variables or responses) that I want to fit using splines (I am open to any choice at the moment). So, I want to create the interpolating function $S(x):{\mathbb R}_+\rightarrow {\mathbb R}$. However, I need to restrict $S(x)$ to satisfy $\frac{d S(x)}{dx} \geq f(x)$, where $f$ is a known continuous function. Is this possible?

Extra: If so, are there any tools in R to do so?