Not sure what you mean by "the derivative" for HSTs. Derivative is a calculus function. Also, the "formulas" quoted only apply to right triangles in which the "legs" are labeled "a" and "b" and the hypotenuse is labeled "c". A more general version would state that the "sum of the squares of the legs is equal to the square of the hypotenuse". I, too, am a retired math and computer science teacher. I spent a lot of years teaching AP calculus to great kids (both in high school and college).