the derivative is a numeric proportion or rate

calculators can specify derivatives by digital numbers

By adding another number to each precise-real number - infinitesmals, rather than complex imaginary-parts - we can have calculators compute derivatives: the second number is the slope at the precise-real number, thus sin(pi/2) = 1.0000000000 , and its second number, its derivative, is 0.0000000000 = cos(pi/2) .

Calculations can follow the derivative chain-rules, eg. for finitesmal a,b and infinitesmals d,e representing their corresponding [cumulated] slopes

- 2(a+d) = (2a)+(2d)
- (a+d)*(b+e) = (ab)+(ae+bd) : (de) being doubly infinitesmal
- sin(a+d) = sin(a)+cos(a)*d : cos(d) being about 1-(d^2/2), and sin(d) about (d)-(d^3/6) - multiply infinitesmally indistinguishable from their first terms
- etc.

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