Limit Definition of a Derivative at a Point
The limit definition of a derivative at a point is a mathematical expression that defines the derivative of a function at a specific point
The limit definition of a derivative at a point is a mathematical expression that defines the derivative of a function at a specific point.
Let’s consider a function f(x) and a point x=a. The derivative of the function at this point, denoted as f'(a) or dy/dx evaluated at x=a, can be defined using the limit as follows:
f'(a) = lim (h→0) [f(a+h) – f(a)] / h
In this definition, h represents a small change in the input variable x. We take the limit of the difference quotient as h approaches zero to find the instantaneous rate of change of the function at the point x=a. This limit is a fundamental concept in calculus and represents the slope of the tangent line to the graph of the function at the point x=a.
To summarize, the limit definition of a derivative at a point is a formula that calculates the derivative by taking the limit of the difference quotient as the interval h approaches zero. It allows us to find the rate of change of a function at a specific point and is a critical concept in calculus and its applications.
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