You don't need induction for (x+x'e)^n+1. The binomial formula can be applied once the arithmetic on dual numbers is introduced.
> We can use this result to prove the same property for any smooth function f. Examining the Taylor expansion of f at zero (also known as its Maclaurin series):
That's not exactly true. For example, arctan = tan^-1 is smooth on R, but its Taylor series only converges for |x|<=1. However, polynomial series expansions exist for |x|>1, where similar arguments would apply. Also, Taylor series and Maclaurin series aren't the same thing. A Maclaurin series is a Taylor series centered at 0.
Lastly, is "differential programming" a new term? It seems weird to me that the machine learning community continues to reinvent terminology for already existing things. What is wrong with automatic differentiation? (Not necessarily a question for the author.) It's not really a new paradigm in the sense of "logic programming", "object-oriented programming", "functional programming", etc., and is instead just a technique.
> We can use this result to prove the same property for any smooth function f. Examining the Taylor expansion of f at zero (also known as its Maclaurin series):
That's not exactly true. For example, arctan = tan^-1 is smooth on R, but its Taylor series only converges for |x|<=1. However, polynomial series expansions exist for |x|>1, where similar arguments would apply. Also, Taylor series and Maclaurin series aren't the same thing. A Maclaurin series is a Taylor series centered at 0.
Lastly, is "differential programming" a new term? It seems weird to me that the machine learning community continues to reinvent terminology for already existing things. What is wrong with automatic differentiation? (Not necessarily a question for the author.) It's not really a new paradigm in the sense of "logic programming", "object-oriented programming", "functional programming", etc., and is instead just a technique.