I want to evaluate an expression of the form $\nabla_\lambda (Y^\eta \partial_\eta Z^\rho)$. 

I'm not sure about how to write down the connection term. My guess is:

$$\nabla_\lambda (Y^\eta \partial_\eta Z^\rho) = \partial_\lambda (Y^\eta \partial_\eta Z^\rho) + Y^\eta \partial_\eta (\Gamma^\rho_{\lambda \alpha} Z^\alpha ) $$

Is this correct?