Is the covariant derivative a tensor?

Question

While studying general relativity, the covariant derivative is constructed (in no rigorous manner) in order to make the derivative of a tensor transform like a tensor. Symbolically, $$ \nabla'_{\mu} V'^{\nu}=\frac{\partial x^{\rho}}{\partial x'^{\mu}} \frac{\partial x'^{\nu}}{\partial x^{\sigma}} \nabla_{\rho} V^{\sigma}.$$ The new derivative is given by $$ \nabla_\mu V^\nu=\partial^{}_\mu V^\nu + \Gamma^\nu_{\mu \rho} V^\rho.$$ The transformation law in the first equation is that of a 2-nd rank tensor. Does that mean the derivative should transform as a $(0,1)$-tensor? If so, I don't see how that is because I don't see how the covariant derivative can be thought of independently without acting on any tensors after it. So I can't write the transformation law for it alone. Does it even make sense to think of it independently in that manner?