Understanding the Limit Definition of a Derivative: A Comprehensive Guide for Calculus Students

Limit Definition of a Derivative at a Point

The limit definition of a derivative at a point is one of the fundamental concepts in calculus

The limit definition of a derivative at a point is one of the fundamental concepts in calculus. It defines the rate at which a function changes at a specific point.

Let’s say we have a function y = f(x), and we want to find the derivative of this function at a specific point x = a. The derivative at this point, denoted as f'(a) or dy/dx evaluated at x = a, gives us the rate of change of the function at that point.

The limit definition of a derivative at a point is given by the following formula:

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

In this formula, h represents a small change in x. As h approaches zero, we are looking at the behavior of the function as it becomes infinitesimally close to the point x = a.

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

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

2. Substitute the function y = f(x) into the formula, replacing f(x) with y.

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

3. Simplify the expression inside the limit by expanding y(a + h) and y(a) using the function definition.

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

4. Cancel out the f(a) terms.

f'(a) = lim(h -> 0) [hf'(a) + higher-order terms] / h

5. Divide both the numerator and denominator by h.

f'(a) = lim(h -> 0) [f'(a) + higher-order terms]

6. Take the limit as h approaches 0.

f'(a) = f'(a) + higher-order terms

7. Simplify the equation by removing the higher-order terms, as they become insignificant when compared to the derivative.

f'(a) = f'(a)

So, we can conclude that the derivative of a function at a specific point a is equal to its derivative at that point.

This limit definition gives us a precise way to calculate the derivative at a point. By taking a smaller and smaller value for h (approaching zero), we obtain a more accurate representation of the instantaneous rate of change of the function at that specific point.

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