I know what a Taylor series involves but you have to know the function; here $\mathcal{L}^*$ is just a function depending on $(v+w)^2$, any kind of function could be inside. How can the $1.11$ approximation be done? and how this generalizes? Does this have a name? $$\mathcal{L}^*_0=\mathcal{L}_0...

Many quantities in physics are defined as ratio of infinitesimal quantities. For example: $$\rho(x)=\frac{dm}{dx}$$ or $$P(t)=\frac{dW}{dt}$$ Are these quantities actually derivatives? I mean if we want to calculate density it wouldn't make sense to select a infinitesimal length element $dx$ and ...