In the definition of Lie derivative
$$ \mathcal{L}_VX^\mu=[V,X]^\mu=V^\mu\partial_\mu X^\nu\partial_\nu-X^\mu\partial_\mu V^\nu\partial_\nu=(V^\mu\partial_\mu X^\nu-X^\mu\partial_\mu V^\nu)\partial_\nu $$
My question is: why in the derivative with respect to $\mu$ coordinate Leibnitz rule is not followed taking into account the change of $\partial_\nu$?
Your use of indices is inconsistent, so I'll rewrite it. Since$$\begin{align}[V,\,X]f&=V^\mu\partial_\mu(X^\nu\partial_\nu f)-V\leftrightarrow X\\&=\color{red}{(V^\mu X^\nu-X^\mu V^\nu)\partial_\mu\partial_\nu f}+\color{limegreen}{(V^\mu\partial_\mu X^\nu-V\leftrightarrow X)\partial_\nu f}\end{align}$$and the red expression is$$V^\mu X^\nu\partial_\mu\partial_\nu f-X^\mu V^\nu\partial_\mu\partial_\nu f=V^\mu X^\nu\partial_\mu\partial_\nu f-X^\nu V^\mu\partial_\nu\partial_\mu f=0$$(by a relabelling and partial derivatives commuting, which is the point @LL's comment made), the green expression gives$$[V,\,X]=(\mathcal{L}_VX^\nu)\partial_\nu,\,\mathcal{L}_VX^\nu=V^\mu\partial_\mu X^\nu-X^\mu\partial_\mu V^\nu.$$