The derivative property of the exponential function can be verified by taking the derivative of the power series definition of the function. Since the power series is a summation, the derivative of each of the terms can be found individually.
Start with the power series definition of the exponential function.
Take the derivative of both sides.
Since the power series is made up of a bunch of summations, the derivative can be applied individually to each term.
Take the derivative of each polynomial.
Applying the identity property of addition we can remove the leading , leaving an expression identical to the original power series definition.
Substituting the exponential function back into the expression we see that the derivative of the exponential function is equal to itself.
This verifies the derivative property of the exponential function.