Derivative Meaning

The derivative of a function 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} =  nx^{n-1}
the derivative of sin(x) is cos(x),
the derivative of f°g is f’°g*g’,
etc.

(3) Logical:
\boxed{\text{f'(x) = d}}
\Updownarrow
\forall \varepsilon, \exists  \delta, \text{ such that }
\boxed{ 0 < |\Delta x| < \delta,  \implies  \Bigr|\frac{f(x+\Delta x)-f(x)}{\Delta x}  - d \Bigr| < \varepsilon }

(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.

(8) The derivative of a real-valued function f in a domain D is the Lagrangian section of the cotangent bundle T*(D) that gives the connection form for the unique flat connection on the trivial R-bundle ßxR for which the graph of f is parallel.

[Source]: Extract from “On Proof and Progess in Mathematics” by William Thurston.

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