Like complex numbers, differentiation can be “pictured” in many ways. The extract below is from the American mathematician William Thurston, Fields Medal winner in 1982 and a pioneer in important areas of topology.
The derivative can be thought of as:
(1) Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.
(2) Symbolic: the derivative of xn is nxn−1, the derivative of sin(x) is cos(x), etc.
(3) Logical: f′(x) = d if and only if for every ε > 0 there is a δ such that when -δ < ∆x < δ, then -ε < [f(x + ∆x) − f(x)]/∆x − d < ε
(4) Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.
(5) Rate: the instantaneous speed of f(t), when t is time.
(6) Approximation: The derivative of a function is the best linear approximation to the function near a point.
(7) Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power.
This is a list of different ways of thinking about or conceiving of the derivative, rather than a list of different logical definitions…
I can remember absorbing each of these concepts as something new and interesting, and spending a good deal of mental time and effort digesting and practising with each, reconciling it with the others. I also remember coming back to revisit these different concepts later with added meaning and understanding.
From “ON PROOF AND PROGRESS IN MATHEMATICS”, BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, Volume 30, Number 2, April 1994, Pages 161-177
Amir Alexander’s book Infinitesimal (there’s a copy in 2B6) is also worth reading on this.