Formal definition of derivative
limit as h approaches 0 of [f(a+h)-f(a)]/h
The derivative of a function is a mathematical concept that represents the rate of change of that function at a specific point. Specifically, the derivative of a function f(x) at a point x=x0, denoted as f'(x0), is defined as the limit of the difference quotient:
f'(x0) = lim h->0 [f(x0 + h) – f(x0)] / h
Intuitively, the derivative gives us information about how quickly the function is changing at that particular point. If the derivative is positive, the function is increasing at that point; if it is negative, the function is decreasing; and if it is zero, the function is not changing at that point. The derivative is an important tool in calculus and is used to solve many problems, such as finding the maximum and minimum values of a function, determining the slope of a tangent line to a curve, and calculating rates of change in real-world applications.
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