${\displaystyle =\sum _{i=0}^{n-1}{{\frac {1}{n}}{\frac {(n-1)^{i}}{n^{i}}}}={\frac {1}{n}}{\frac {1-{({\frac {n-1}{n}}})^{n}}{1-{\frac {n-1}{n}}}}=1-({\frac {n-1}{n}})^{n}}$

The derivative of a real-valued function f in a domain D is the Lagrangian section of the cotangent bundle T*(D) that gives the connection form for the unique flat connection on the trivial R-bundle D×R for which the graph of f is parallel.