Limit Definition of Derivative
limit (as h approaches 0)= F(x+h)-F(x)/h
The limit definition of derivative is a mathematical formula used to calculate the instantaneous rate of change of a function at a specific point. It is denoted by f'(x) and is defined as the limit of the difference quotient as h approaches zero. The difference quotient is the expression (f(x+h) – f(x))/ h. In other words, the derivative of a function f(x) at a point x is the slope of the tangent line to the graph of f(x) at that point x.
The limit definition of the derivative can be written mathematically as:
f'(x) = lim(h → 0) [f(x+h) – f(x)] / h
where f(x) is the original function and f'(x) is its derivative at a specific point x. The h in the formula represents the change in x or the distance between x and a nearby point on the graph. As h approaches zero, the difference quotient becomes closer and closer to the slope of the tangent line to the graph of f(x) at x.
The limit definition of the derivative is an important concept in calculus and is used in many calculations involving rates of change and optimization problems. It provides a way to calculate the instantaneous rate of change of a function, which can be used to determine the slope of a function at any given point.
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