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

(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)