Definition of the Derivativef'(x) = ___________________
lim h->0 f(x+h)-f(x) / h
The derivative of a function f(x) at a point x gives the instantaneous rate of change of the function at that point. It is the limit of the ratio of the change in the function value and the change in the input variable as the change in the input variable approaches zero. Mathematically, the derivative of f(x) with respect to x is denoted by f'(x) and is defined as:
f'(x) = lim delta x -> 0 [f(x + delta x) – f(x)] / delta x
This expression represents the slope of a tangent line to the function f(x) at the point x. In words, it tells us how fast the function is changing at that point. The derivative is a fundamental concept in calculus and has many important applications in mathematics, science, engineering, and economics.
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