Deconstructing the numerator in derivatives I'm trying to deconstruct $\frac{d(\sqrt{x^2+y^2})}{dt}$ into possibly $\frac{dx}{dt}$ and $\frac{dy}{dt}$, but I have no idea where to start or what to search.

The chain rule says $\frac{d f(g(t))}{dt}=f'(g(t))g'(t)$.  In this case, take $f(z)=\sqrt z$ and $g(t)=x(t)^2+y(t)^2$ so that
$$\frac{d\sqrt{x(t)^2+y(t)^2}}{dt}=\frac1{2\sqrt{x(t)^2+y(t)^2}}\cdot\frac d{dt}(x(t)^2+y(t)^2)$$
and the rest follows pretty quickly with another application of the chain rule.  I would hesitate to call $f$ a numerator, I'd say it's the function which your taking the derivative of, not sure if there's a slick term like "integrand".
