Understanding the Limit Definition of a Derivative at a Point | Calculating the Instantaneous Rate of Change of a Function

Limit Definition of a Derivative at a Point

The limit definition of a derivative at a point is an expression used to calculate the instantaneous rate of change of a function at a specific point

The limit definition of a derivative at a point is an expression used to calculate the instantaneous rate of change of a function at a specific point. It is denoted as follows:

\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) – f(a)}{h} \]

In this expression, \( f'(a) \) represents the derivative of the function \( f(x) \) at the point \( a \). The limit of the fraction on the right side is taken as the variable \( h \) approaches zero.

To understand this concept better, let’s break down the terms:

1. \( f(a + h) \): It represents the value of the function \( f(x) \) at \( a + h \), i.e., the value of the function at a point close to \( a \).

2. \( f(a) \): It represents the value of the function \( f(x) \) at \( a \), i.e., the value of the function at the specific point we are interested in finding the derivative of.

3. \( \frac{f(a + h) – f(a)}{h} \): It calculates the average rate of change of \( f(x) \) over the interval from \( a \) to \( a + h \). This is done by finding the difference in values between the two points and dividing it by the difference in their corresponding x-coordinates. Taking the limit as \( h \) approaches zero ensures that we are calculating the instantaneous rate of change at \( a \).

In simpler terms, the limit definition of a derivative at a point measures how the function changes at an infinitesimally small interval around that point. It determines the slope of the tangent line to the function at the given point.

By evaluating this limit, we can find the derivative of a function at a specific point, which gives us valuable information about its behavior and allows us to analyze its rate of change with high precision.

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