How obtain derivative at a point if we know the derivative at another different point

Question

Why it is sufficient to compute the derivative of a function f(t) at the point t=0 to obtain the derivative at a point t'>0 ? Is there a relation between the two derivatives at such points?

Answer 1

It's not sufficient. For instance, the functions $f(t) = t^2$ and $g(t) = t^3$ both have derivative zero at $t = 0$, but their derivatives (and values) elsewhere are rather different.