This is from a set of lecture notes on differential geometry.
We make the notational definition: $$D_{\mu}\equiv D_{\partial_\nu} \tag{1}$$
The covariant derivative is defined by its action on basis vectors:
$$D_\mu(\partial_\nu)=\Gamma^\lambda_{\mu\nu}\partial_\lambda, \tag{2}$$
where I have used the summation convention and suppressed the evaluation at the point $p$ in the derivative operators $(\partial_\lambda|_p\rightarrow\partial_\lambda)$.
We then proceed to define the covariant derivative of $Y$ along $X$: $$\begin{align} D_XY&=X^\mu D_\mu(Y^\nu\partial_\nu) \\ &= X^\mu(\partial_\mu(Y^\nu)\partial_\nu+Y^\nu D_\mu(\partial_\nu)) \\ &= (X^\mu\partial_\mu Y^\lambda+\Gamma^\lambda_{\mu\nu}X^\mu X^\nu)\partial_\lambda \end{align}.$$
It is in the second line I am getting lost, we apply the covariant derivative to $Y^\mu$ and then to $\partial_\nu$ as with any derivative operator, but the connection coefficients do not appear in the first term of the second line?
I am a physics student so I have seen the covariant derivative written out, but this is a more mathematical text and I am only now seeing the covariant derivative defined more carefully.
Any help appreciated, thanks.
A connection $\nabla$ is indeed asked to check the Leibniz rule $$\nabla_X(fV)=(Xf)V+f(\nabla_XV),$$ where $f$ is a scalar function and $V$ a vector field. The first term of the sum does not depend of the connection, so there won't be Christoffel coefficients implied there.