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 describes how the function changes at a specific point. More formally, the derivative of a function f(x) at a point x = a is defined as the limit of the slope of a secant line passing through the point (a, f(a)) and a nearby point (a+h, f(a+h)) as h approaches zero. The formula is written as:
f'(a) = lim h→0 [(f(a+h) – f(a))/h]
The result of this limit is the instantaneous rate of change of the function at the point x=a, which gives the slope of the tangent line to the curve at that point. The derivative of a function measures how much the value of the function changes when the input is changed by a very small amount, and it is a fundamental concept in calculus used to solve a wide range of problems in science, engineering, economics, and other fields.
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