Understanding the Alternate Definition of Derivative | Exploring the Intuitive Concept of Rate of Change

Alternate definition of derivative

The alternate definition of derivative is also known as the incremental definition

The alternate definition of derivative is also known as the incremental definition. It provides an intuitive understanding of the derivative as the rate of change or slope of a function at a given point.

To explain the alternate definition, let’s consider a function f(x) and a specific point on the graph of the function, say (a, f(a)). The derivative of f(x) at x = a is defined as:

f'(a) = lim (h->0) [f(a + h) – f(a)] / h

In this definition, h represents a small increment in the x-values around the point a. By taking the limit as h approaches 0, we are essentially considering smaller and smaller intervals around the point to calculate the rate of change or slope.

To compute the derivative using the alternate definition, follow these steps:

1. Choose a specific value for a.
2. Select a small increment h.
3. Compute f(a + h) and f(a).
4. Take the difference between f(a + h) and f(a) and divide it by h.
5. Calculate the limit as h approaches 0.

The alternate definition allows us to find the derivative of a function at a specific point without relying on any explicit formula. It is particularly useful when functions are not given explicitly, but instead defined by a set of data or in terms of other functions.

However, in practice, the alternate definition is less commonly used than other derivative rules, such as the power rule, product rule, or chain rule, which provide more efficient methods for finding derivatives in most cases.

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