How to Derive Christoffel Symbols and the Covariant Derivative

Here is my new and improved derivation of Christoffel symbols and the covariant derivative. We begin with the metric. Let's convert the rank-one tensors (xixj) to x^2 and pull it out of the radical:

Next, let's take the ordinary derivative, using the product rule and chain rule of calculus:

In the last equation above, we divided both sides of the equation by (gij)^.5. Below we use identities and substitutions to put the equation into a covariant derivative format, which includes the Christoffel symbol:

Finally, we use a similar process to derive the covariant derivative and Christoffel symbol for a contra-variant metric tensor and co-variant rank-one tensor. These tensors are the inverse of the tensors we worked with above (co-variant metric tensor and contra-variant rank-one tensor).