A derivative is formally defined through a limit, a precise mathematical procedure describing how a function's output changes as its input changes by an infinitesimally small amount. Working through that full limit definition by hand every single time a derivative is needed would be extremely slow and impractical for anything beyond the simplest functions, which is exactly why calculus develops a set of differentiation rules, the power rule, product rule, chain rule, among others, that let a derivative be calculated quickly through a mechanical, memorised procedure instead.
Each rule was itself proven once from the limit definition, then reused freely
Every differentiation rule is itself derived, at some point, directly from the formal limit definition of a derivative, applied to a specific general pattern of function. The power rule, for instance, is proven once, generally, for any function raised to a power, and once that proof is established, the rule can be applied directly to any specific function matching that pattern without redoing the underlying limit calculation each time. This is the core efficiency differentiation rules provide: the hard foundational work of connecting a specific pattern back to the limit definition only has to be done once, and the resulting rule can then be reused indefinitely afterward.
Combining several rules handles even quite complicated functions quickly
Real functions a student or working mathematician actually needs to differentiate are often combinations of several simpler patterns at once, a product of two functions, or a function nested inside another function, and differentiation rules are specifically built to handle exactly this kind of combination. The product rule handles a product of two functions, the chain rule handles a function nested inside another, and by combining the appropriate rules together in sequence, even a fairly complicated function's derivative can typically be calculated quickly and mechanically, without ever needing to return to the formal limit definition for that specific case.
What we're still unsure about
The differentiation rules themselves, and their formal derivation from the limit definition of a derivative, are precisely defined, fully settled mathematics with no genuine ambiguity, confirmed through rigorous mathematical proof. What varies is more a matter of pedagogy than open mathematical questions: maths educators continue to debate exactly how much time students should spend proving each rule from first principles, using the formal limit definition, versus simply learning to apply the already-proven rules efficiently, without one universally agreed answer on the right balance for a given course level.
This sits inside Derivatives & Differentiation Rules, one of eight topics in Calculus, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.