Suppose that $ (U,x^1,x^2,...,x^n) $ and $ (V,y^1,y^2,...,y^n) $ are two coordinate charts on a manifold.Then $$ {\partial \over \partial x^j}=\sum_i {\partial y^i \over \partial x^j } {\partial \over \partial y^i }. $$
question Why $ \sum_i {\partial y^i \over \partial x^j } {\partial x^j\over \partial y^i } $=1? I think it is n.
Hint: if $T$ is a transformation from $U$-coordinates to $V$-coordinates and $T^{-1}$ is its inverse, then $T T^{-1} = {\rm Id} = T^{-1} T$. So, the ${\rm D} (TT^{-1}) = {\rm D}({\rm Id}) = {\rm Id}$. Can you proceed further from here?