The Derivative Of A Function: A Fundamental Concept In Calculus & Its Practical Applications

Definition of the Derivativef'(x) = ___________________

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

The derivative of a function f(x) is defined as the rate of change of the function at a specific point x. It is denoted by f'(x) and can be mathematically expressed as the limit of the ratio of the change in the function output to the change in the function input, as the change in input approaches zero. In symbols, we can write:

f'(x) = lim(h→0) [(f(x + h) – f(x)) / h]

This limit represents the instantaneous rate of change of the function at point x. It gives us information about the slope of the tangent line to the graph of the function at that point, which is a crucial concept in calculus and is used in many applications, including physics, engineering, economics, and more.

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