Covariant derivative in cylindrical coordinates I am confused by the Wolfram article on cylindrical coordinates. Specifically, I do not understand how they go from equation (48) to equations (49)-(57).
Equation (48) shows that the covariant derivative is:
$$A_{j;k} = \frac{1}{g_{kk}}\frac{\partial A_j}{\partial x_k} - \Gamma^i_{jk}A_i$$
The next few equations expand this for the case of cylindrical coordinates, equation (50) is:
$$A_{r;\theta} = \frac{1}{r}\frac{\partial A_r}{\partial \theta} - \frac{A_\theta}{r}$$
The contravariant metric tensor has non-zero elements:
$$g^{11} = 1$$
$$g^{22} = \frac{1}{r^2}$$
$$g^{33} = 1$$
And the Christoffel symbols of the second kind have non-zero elements:
$$\Gamma^1_{22} = -r$$
$$\Gamma^2_{12} = \frac{1}{r}$$
$$\Gamma^2_{21} = \frac{1}{r}$$
If I plug these values back into their definition of the covariant derivative I get for equation (50):
$$A_{r;\theta} = \frac{1}{r^2}\frac{\partial A_r}{\partial \theta} - \frac{A_\theta}{r}$$
Why does this not match up with their results?

The thing you forgot is the scale factor $\frac{1}{r}$ given in equation (14).  See Scale Factor in Mathworld.
