I am reading Ray d'Inverno's book, 'Introducing Einstein's Relativity' and there is a particular derivation of the geodesic equation that I get stumped on (chapter 7). It is a variational method and the final equation is
where f is the Lagrangian, x_alpha, beta etc are the coordinates
and _dot denotes differenation with respect to an affine parameter.
Now you can carry this computation and derive the equivalent equation but using christoffel symbols. But in this derivation, one line is
This equation is a sum over all indices (alpha, beta etc).
My question is shouldn't this last term be differentiated with the product rule, so that
f is a summation over all indices (ALPHA INCLUDED), so do we not need extra terms to determine these if we are differentiating with respect to alpha??
Why can we jump to just
Please help. I feel I'm missing a point here.