Given a smooth manifold $N$ of 4 dimension. By smooth manifold here, we mean that $N$ is a manifold with a smooth atlas $\,\mathcal A=\Big\{\big(U_i,\varphi_i:\, U_i\to\mathbb R^4\big) \big|\,i\in I \Big\} $.
Let \begin{align} x:\ N&\longrightarrow\mathbb R^4 \\ p&\longmapsto x(p)=\big(x_1,x_2,x_3,x_4\big)(p) \\ T:\ \mathbb R^4&\longrightarrow\mathbb R^4 \\ (x_1,x_2,x_3,x_4)&\longmapsto T(x_1,x_2,x_3,x_4) \end{align} be diffeomorphisms. Then the map \begin{align} (T\circ x)^{-1}:\ \mathbb R^4&\longrightarrow N \\ (u_1,u_2,u_3,u_4)&\longmapsto x^{-1}\circ T^{-1}(u) \end{align} is a diffeormorphism and \begin{align} \frac{\partial\big(T\circ x\big)^{-1} }{\partial u_1}(u),\,\frac{\partial\big(T\circ x\big)^{-1} }{\partial u_2}(u),\,\frac{\partial\big(T\circ x\big)^{-1} }{\partial u_3}(u),\,\frac{\partial\big(T\circ x\big)^{-1} }{\partial u_4}(u) \end{align} is a basis for $T_pN$ where $\,p=x^{-1}\circ T^{-1}(u)$.
My question is how to calculate the partial derrivative $\,\displaystyle\frac{\partial\big(T\circ x\big)^{-1} }{\partial u_i}(u)\,$ in this problem ? I hope anyone could help me with this. Thanks.