I am studying Moroianu book lectures on Kähler geometry. I am trying to show that the map $df_x$ is well defined. Choose charts $\phi^{\prime}_{U^{\prime}}$ and $\psi^{\prime}_{V^{\prime}}$ around x and f(x). It follows that $[U,u] = [U^{\prime},u^{\prime}]$ and $[V,v] = [V^{\prime},v^{\prime}]$. It follows $$u = d\phi_{{UU^{\prime}}_{\phi_U^{\prime}(x)}} (u^{\prime})$$ $$v = d\psi_{{VV^{\prime}}_{\psi_V^{\prime}(x)}} (v^{\prime})$$ Therefore $df([U,x]) = [V,d(\psi_V \circ f \circ \phi^{-1}(U) \circ d\phi_{{UU^{\prime}}_{\phi_U^{\prime}(x)}} (u^{\prime})] = [V, d(\psi_V \circ f \circ \phi_U^{\prime})x]$
I am missing $\psi_V^{\prime}$. I am not sure how to get rid of $\psi_V$? In order to show well-definedness.
Extending my comment, you have by definition $$ \begin{align} \mathrm{d}f_x([U,u]) = [V, \mathrm{d}(\psi_{V}\circ f \circ \phi_{U}^{-1})_{\phi_U(x)}(u)]. \tag{1} \end{align} $$ The definition of tangent vectors as equivalence classes means that for tangent vectors at $x\in M$ $$ \begin{align} [U^{\prime},u^{\prime}]=[U,\mathrm{d}(\phi_{UU^{\prime}})_{\phi_{U^{\prime}}(x)}(u^{\prime})] \tag{2} \end{align} $$ and for tangent vectors at $f(x)\in N$ $$ \begin{align} [V^{\prime},v^{\prime}]=[V,\mathrm{d}(\psi_{VV^{\prime}})_{\psi_{V^{\prime}}(f(x))}(v^{\prime})] \tag{3} \end{align} $$ where $\phi_{U^{\prime}U}=\phi_{U^{\prime}}\circ \phi_{U}^{-1}$ and $\psi_{V^{\prime}V}=\psi_{V^{\prime}}\circ \psi_{V}^{-1}$ are the transition maps between two charts around $x\in M$ and $f(x)\in N$.
Carefully unraveling all the definitions you now get $$ \begin{align} \mathrm{d}f_x([U^{\prime},u^{\prime}]) &= \mathrm{d}f_x\big([U,\mathrm{d}(\phi_{UU^{\prime}})_{\phi_{U^{\prime}}(x)}(u^{\prime})]\big) \\ &= \big[V, \mathrm{d}(\psi_{V}\circ f \circ \phi_{U}^{-1})_{\phi_{U}(x)}\circ \mathrm{d}(\phi_{UU^{\prime}})_{\phi_{U^{\prime}}(x)}(u^{\prime})\big] \\ &= \big[V, \mathrm{d}\big(\psi_{V}\circ f \circ \phi_{U}^{-1} \circ \phi_{UU^{\prime}}\big)_{\phi_{U^{\prime}}(x)}(u^{\prime})\big] \\ &= \big[V, \mathrm{d}\big(\psi_{V} \circ \psi_{V^{\prime}}^{-1} \circ \psi_{V^{\prime}} \circ f \circ \phi_{U}^{-1} \circ \phi_{U} \circ \phi_{U^{\prime}}^{-1}\big)_{\phi_{U^{\prime}}(x)}(u^{\prime})\big] \\ &= \big[V, \mathrm{d}(\psi_{VV^{\prime}})_{\psi_{V^{\prime}}(f(x))} \Big(\mathrm{d}\big(\psi_{V^{\prime}} \circ f \circ \phi_{U^{\prime}}^{-1}\big)_{\phi_{U^{\prime}}(x)}(u^{\prime})\Big)\big] \\ &= \big[V^{\prime}, \mathrm{d}\big(\psi_{V^{\prime}} \circ f \circ \phi_{U^{\prime}}^{-1}\big)_{\phi_{U^{\prime}}(x)}(u^{\prime})\big] \end{align} $$ where we used in order of appearance eq. (2), eq. (1), chain rule, definition of transition maps, chain rule, eq. (3).