Definition of derivative
web.iit.edu › pdfs › what_are_DerivativeDerivative as a function •As we saw in the answer in the previous slide, the derivative of a function is, in general, also a function. •This derivative function can be thought of as a function that gives the value of the slope at any value of x. •This method of using the limit of the difference quotient is also
Derivative - Wikipedia
https://en.wikipedia.org/wiki/DerivativeRelative to a hyperreal extension R ⊂ R of the real numbers, the derivative of a real function y = f(x) at a real point x can be defined as the shadow of the quotient ∆y/∆x for infinitesimal ∆x, where ∆y = f(x + ∆x) − f(x). Here the natural extension of f to the hyperreals is still denoted f. Here the derivative is said to exist if the shadow is independent of the infinitesimal chosen.