Let $(x_i, y_i)$ be a finite sample of points. I want to find the slope of the tangent at $x$ to a curve "best fit" to the given data. In my problem area, (see context below), the simplest of techniques is to perform an $L^2$ minimization to find the closest parabola to the curve. Then its derivative at the point $x$ can be computed. There are various models that suggest different family of curves from which a best one can be chosen to fit the data as well.
My question is: Suppose I only assume that the sample points satisfy $$y_i = f(x_i) + \epsilon_i$$ where $f$ is smooth and $\epsilon_i$ are iid normal random variables. Are there any techniques for finding a best function $f$, or really just the derivative of $f$ at some point $x$ without using any model related to the underlying problem. I am thinking of something like a "minimum entropy curve".
The above picture is the general situation. A series of point that lie close to a curve that has "very little variation". Perhaps minimizing both the $L^2$ distance, and the energy of the curve is the proper technique.
Context: This problem arises in Math Finance but I believe it is more general. In that context though it is the issue of finding the at the money skew to the implied volatility curve.