having a little difficulty with this conceptually. Can someone quickly walk through this differentiation please?
$K=\frac{1}{2}g_{ab}\dot{x}^{a}\dot{x}^{b}$
Find $\frac{dK}{dx^{a}}$
In this case, $x^{a}$ is a function of $t$, and the differentiated $x$'s in $K$ are with respect to $t$.
$g_{ab}$ is the metric tensor shouldn't really matter I think.
We have $$\frac{\partial K}{\partial x^k}=\frac{1}{2}\frac{\partial g_{ab}}{\partial x^k}\dot{x}^a\dot{x}^b.$$