Deriving an Operator From the Chain Rule

Suppose a function  
\[f\]
  is defined in terms of variables  
\[x_1, \l x_2,..., \; x_n\]
  that are themselves functions of  
\[t\]
. Then using the Differentiation - The Chain Rule we can write
\[\frac{df}{dt}= \frac{dx_1}{dt}\frac{\partial f}{\partial x_1} + \frac{dx_2}{dt}\frac{\partial f}{\partial x_2} +...+ \frac{dx_n}{dt}\frac{\partial f}{\partial x_n} \]
.
\[\frac{df}{dt}= \frac{dx_1}{dt}\frac{\partial }{\partial x_1} + \frac{dx_2}{dt}\frac{\partial }{\partial x_2} +...+ \frac{dx_n}{dt}\frac{\partial }{\partial x_n} \]
.
Ising operatprs like this we can build an Operator Algebra.

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