The Limit Definition of Derivative | Calculating the Rate of Change with Precision

Alternate definition of derivative

The alternate definition of derivative is known as the limit definition of derivative

The alternate definition of derivative is known as the limit definition of derivative. This definition describes how the derivative of a function can be calculated by taking the limit of a difference quotient.

Let’s consider a function f(x). The derivative of f(x) at a specific point x=a can be defined 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 around the point a. By taking the difference between the function values at the points a+h and a and dividing it by h, we are determining the average rate of change of the function over a small interval. As h approaches zero, this average rate of change becomes the instantaneous rate of change, which is the derivative at the point x=a.

By applying the limit process, we can find the slope of the tangent line to the graph of the function at the point a. This slope represents the rate at which the function is changing at that specific point.

The alternate definition of derivative is particularly useful when the function is not expressed in a specific formula or equation but is instead given as a set of data points or a graph. In such cases, we can estimate the derivative by calculating the difference quotient for small intervals around the point of interest.

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