The limit definition of the derivative, through the limit as (h) approaches 0, determines the instantaneous rate of change of a function at a specific point on its curve.
Explaining the Limit Definition of the Derivative:
In calculus, the limit definition of the derivative is a fundamental concept that determines the instantaneous rate of change of a function at a specific point. The derivative of a function (f(x)) at a point (x=a) is defined as:
![\[f'(a) = \lim_(h \to 0) (f(a + h) - f(a))/(h).\]](https://img.qammunity.org/2024/formulas/mathematics/college/56w3byeu8x2jorfugj0jm72k1tl2czu1rn.png)
This formula represents the slope of the tangent line to the curve at the point (x=a). As (h) approaches 0, the secant line becomes a tangent line, and the limit provides the instantaneous rate of change at that precise moment. The derivative, (f'(x)), expresses the rate at which (f(x)) changes concerning (x).
By analyzing the limit definition, one gains insight into how small changes in (x) correspond to changes in (f(x)). This concept is crucial for understanding motion, optimization, and various applications in science and engineering.