In Thurston's philosophical paper, "On Proof and Progress in Mathematics", Thurston points out that mathematicians often think of a single piece of mathematics in many different ways. As an example, he singles out the concept of the derivative of a function, and gives seven different elementary ways of thinking about it:

(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 $\cos(x)$, the derivative of $f \circ g$ is $f' \circ g * g'$, etc.

(3) Logical: $f'(x) = d$ if and only if for every $\epsilon$ there is a $\delta$ such that when $0 \lt | \Delta x | \lt \delta, |\frac{f(x + \Delta x) - f(x)}{\Delta x} - d| \lt \delta$

(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.

Thurston goes on to state that "The list continues; there is no reason for it ever to stop," so let's keep it rolling! Can you come up with other ways of thinking about the derivative? I should remark that this is a list of different ways of thinking about the derivative, which isn't the same thing as a list of different formal definitions of the derivative. Remember to limit yourself to one answer per post.

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