How can I prove the following relation from tensor calculus?

Question

$\frac{\partial \bar{x}_{i}}{\partial x_{r}} \frac{\partial {x}_{r}}{\partial \bar{ x_{j}}} = \delta^i_j \quad (The \quad Kronecker \quad Delta) \quad \quad\quad $ $\rightarrow ( \text{In my attempt, I ran r over {1,2,3..,n}}) \rightarrow \frac{\partial \bar{x}_{i}}{\partial x_{1}} \frac{\partial {x}_{1}}{\partial \bar{ x_{j}}} + \frac{\partial \bar{x}_{i}}{\partial x_{2}} \frac{\partial {x}_{2}}{\partial \bar{ x_{j}}} + ...+ \frac{\partial \bar{x}_{i}}{\partial x_{n}} \frac{\partial {x}_{n}}{\partial \bar{ x_{j}}} = \delta^i_j + \delta^i_j + ... +\delta^i_j = n \delta^i_j \quad \quad \text{But I know that it can not be right.} $

Answer 1

You need to use the chain rule backwards. $\frac{\partial \bar{x}_{i}}{\partial x_{r}} \frac{\partial {x}_{r}}{\partial \bar{ x_{j}}} = \frac{\partial \bar{x}_{i}}{\partial \bar{x}_j} = \delta^i_j $