Limit-free derivatives

Algebraic double roots are used by [1] to motivate the following limit-free definition ofderivative:‘A functionf(x)has a derivativematx = aiffor some valuec.’As we shall see later, ‘function’ in this definition will actually be restricted to real polynomials and [1] concludes‘We have shown how an elementary algebraic principle — double roots — can lead to a complete calculus of polynomials and related functions, without the need for a limit concept.</jats:p