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 x^{n} is nx^{n−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.

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