If I have some rank-2 tensor $g_{ab}$ with components dependent on some coordinate system $x^a$, how do I do the following differentiation in index notation (assuming the $\dot x^d$ are independent of $x^d$)?
$$\frac{\partial (g_{ab}^\frac12 \dot x^a \dot x^b)}{\partial x^c} = \frac{1}{2} g_{ab}^\frac {-1}{2} \frac{\partial g_{ab}}{\partial x^c}\dot x^a \dot x^b $$
Clearly this cannot be correct as the dummy indices a and b not occur three times on the right-hand side. Could somebody explain how to do this differentiation while remaining in index notation (if possible)?
Thanks in advance.
Take $F^2=g_{ab}\dot{x}^a\dot{x}^b$.
So $2F\frac{\partial F}{\partial x^k}=\frac{\partial g_{ab}}{\partial x^k}\dot{x}^a\dot{x}^b$.
Hence $\frac{\partial F}{\partial x^k}=\frac{1}{2F}\frac{\partial g_{ab}}{\partial x^k}\dot{x}^a\dot{x}^b$