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

Alternate Definition of Derivative

The derivative of a function represents the rate at which the function changes at a specific point

The derivative of a function represents the rate at which the function changes at a specific point. Most commonly, the derivative is defined as the slope of the tangent line to the graph of the function at that point. However, there is an alternate definition of the derivative known as the limit definition of the derivative.

The limit definition of the derivative states that for a function f(x), the derivative of f(x) at a specific point ‘a’ is given by:

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

Here, ‘h’ represents a small change in the value of x. By taking the limit as h approaches 0, we are essentially studying the behavior of the function as these small changes become infinitesimally small.

To find the derivative using this definition, we follow the steps:

1. Choose a specific value of ‘a’ for which we want to find the derivative.
2. Calculate f(a + h) by substituting ‘a + h’ into the function.
3. Subtract f(a) from f(a + h) to find the change in the function value.
4. Divide the change in the function value by ‘h’ to find the average rate of change.
5. Take the limit as h approaches 0 to obtain the instantaneous rate of change, which is the derivative.

Let’s illustrate this with an example:

Consider the function f(x) = x^2. We want to find the derivative at x = 2 using the alternate definition.

Step 1: Choose ‘a’ as 2.
Step 2: f(a + h) = f(2 + h) = (2 + h)^2 = 4 + 4h + h^2.
Step 3: f(a + h) – f(a) = (4 + 4h + h^2) – (2^2) = 4h + h^2.
Step 4: [f(a + h) – f(a)] / h = (4h + h^2) / h = 4 + h.
Step 5: Taking the limit as h approaches 0, we have lim(h -> 0) (4 + h) = 4.

Therefore, the derivative of f(x) = x^2 at x = 2 is 4.

The alternate definition of the derivative provides a rigorous way to calculate derivatives using the concept of limits. It can be particularly useful when dealing with more complex functions where the slope of the tangent line is not straightforward to find.

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