Understanding the Derivative: Instantaneous Rate of Change and Slope of Functions

Definition of the Derivativef'(x) = ___________________

lim h->0 f(x+h)-f(x) / h

The derivative f'(x) of a function f(x) represents the instantaneous rate of change or slope of the function at a particular point x. It is defined as the limit of the difference quotient Δy/Δx as Δx approaches zero:

f'(x) = lim(Δx -> 0) [f(x + Δx) – f(x)] / Δx

Alternatively, the derivative can be defined as the slope of the tangent line to the curve at a particular point.

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