The New Calculus

This looks like the work of a crank.

Yes, ‘Calculus’ as it was classically first derived (and as it is taught at high school level) is based on infinitesimals and those are demonstrably not rigorous, famously leading to Bishop Berkley dismissing them as “ghosts of departed quantities”.

However Cauchy and Weierstrass laid down the rigorous “ε & δ” limits-based formulation several hundred years ago, and ushered in the eras of Real and subsequently Complex Analysis.