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For instance, if the inverse of a differentiable function is differentiable this would be easy to prove since

$$\frac{\partial(x_1\ldots x_n)}{\partial(y_1\ldots y_n)}\,\frac{\partial(y_1\ldots y_n)}{\partial(x_1\ldots x_n)}=1\neq 0$$

However I don't know whether this is correct or not.