Is the derivative function typically "worse" than the original function?

Question

For instance, the absolute value function is defined and continuous on the whole real line, but its derivative behaves like a step function with a jump-discontinuity. For some nice functions, though, such as $e^x$ or $\\sin(x)$, the derivatives of course are no "worse" than the original function. Can I say something that is *typical* of the derivative? Is it typically not as nice as the original function?

Answer 1

for integer polynomial functions, the derivative gets "better"

the derivative reduces the degree by one each time

take enough derivatives and the polynomial "disappears" completely and becomes zero

f(x) =ax^3 + bx^2 + cx + d

f'(x) =3ax^2 +2bx +c

f"(x) = 6ax +2b

f"'(x) = 6a

f'^v(x) = 0

f^{vi(x) = 0}

^{all further derivatives also = 0}