What is the proper way to take the first order derivatives when creating a Christoffel symbol?

Question

I'm trying to follow the construction of a Christoffel symbol on this link Einstein Relativity Easy. Specifically, I'm trying to come up with a factory (simple, unambiguous steps) for generating the symbol rather than analyzing every step. Halfway down the page, it's time to compute $\frac{\partial r}{\partial x}$. $$\frac{\partial r}{\partial x}=\frac{1}{2}\frac{2x}{\sqrt{x^2+y^2}}=\frac{x}{r}=Cos(\theta)$$ It's obvious here that the author is making use of the formula:$$r(x,y)=\sqrt{x^2+y^2}$$The problem I have is that there's another way to obtain $\frac{\partial r}{\partial x}$: $$r(x,\theta)=\frac{x}{Cos(\theta)}$$$$\frac{\partial r}{\partial x}=Sec(\theta)$$ Why is the first formula preferred over the second formula?