The key idea of Calculus at level 7 is that there is a link in both directions between the gradient of a function and the original function.

This relation between the function and the gradient function can be exploited in many applications. Indeed it might be argued that this relation was one of the most significant advances in mathematics ever. This is because that a great number of practical situations can be modelled and solved by calculus techniques.

In the case of polynomials, which dominate this thread, there is a very simple relation between function and gradient function and hence between function and derivative, and function and integral. This relation can be determined using a first principles approach that can be demonstrated to, and used by, students at this level.

This key idea develops from almost all of the non-statistical material at previous levels as well as Equations and expressions and Patterns and relationships at this level. This is particularly true of all the algebra in levels up to and including this one.

This key idea is extended in the key idea of Calculus at level 8 where more of the theoretical bases of Calculus are investigated and more applications are considered partly because the functions on which students can operate extend beyond polynomials. Some of these applications are based on forming and solving equations that involve derivatives.