Understanding the Limit Definition of Derivative: A Rigorous Approach to Calculating Instantaneous Rate of Change

Limit Definition of Derivative

The limit definition of derivative is the mathematical definition that describes the rate of change of a function at a specific point

The limit definition of derivative is the mathematical definition that describes the rate of change of a function at a specific point. It provides a precise way to calculate the instantaneous rate of change, or the slope of a curve, at a particular point.

Let’s consider a function f(x) and choose a specific point a. The derivative of f(x) at x = a is defined as follows:

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

In this definition, h represents a small increment in x. By taking the limit as h approaches 0, we ensure that the two points (a, f(a)) and (a+h, f(a+h)) on the function become arbitrarily close to each other, resulting in a more accurate approximation of the instantaneous rate of change at a.

To calculate the derivative using this definition, follow these steps:

1. Start with the limit definition: f'(a) = lim(h->0) [f(a + h) – f(a)] / h.

2. Substitute the function f(x) with the specific function you are working with. For example, if f(x) = x^2, then the equation becomes:
f'(a) = lim(h->0) [(a + h)^2 – a^2] / h.

3. Expand and simplify the expression inside the limit:
f'(a) = lim(h->0) [a^2 + 2ah + h^2 – a^2] / h
= lim(h->0) [2ah + h^2] / h.

4. Cancel out the h terms:
f'(a) = lim(h->0) 2a + h.

5. Take the limit as h approaches 0:
f'(a) = 2a.

So, for the function f(x) = x^2, the derivative of f(x) is 2x. This means that at any point a on the graph of f(x) = x^2, the slope or rate of change of the curve is equal to 2a.

The limit definition of derivative offers a rigorous way to find the slope of a curve at a specific point, and it forms the foundation for further study of calculus and its applications in various fields.

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