Derivative Of F(x) = 3/x Using First Principles A Step-by-Step Guide

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Introduction

In calculus, the derivative of a function, denoted as f'(x), represents the instantaneous rate of change of the function with respect to its variable. One fundamental way to calculate the derivative is by using the first principles, also known as the definition of the derivative. This method involves evaluating a limit that represents the slope of the tangent line to the function's graph at a particular point. This article will delve into the process of finding the derivative of the function f(x) = 3/x using first principles, providing a step-by-step explanation and illuminating the underlying concepts. Understanding the first principles is crucial for grasping the essence of differentiation and its applications in various fields like physics, engineering, and economics.

What are First Principles?

The derivative from first principles is defined as follows:

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

This formula calculates the limit of the difference quotient as h approaches zero. The difference quotient, [f(x + h) - f(x)] / h, represents the average rate of change of the function over a small interval of length h. As h gets infinitesimally small, this average rate of change approaches the instantaneous rate of change, which is the derivative. Mastering this fundamental concept allows one to tackle more complex differentiation problems and appreciate the elegance of calculus.

Steps to Apply First Principles

  1. Evaluate f(x + h): Substitute x + h into the function f(x).
  2. Calculate f(x + h) - f(x): Find the difference between the function evaluated at x + h and the function evaluated at x.
  3. Divide by h: Divide the difference obtained in the previous step by h.
  4. Evaluate the limit as h approaches 0: Find the limit of the expression obtained in the previous step as h approaches zero.

Applying First Principles to f(x) = 3/x

Step 1: Evaluate f(x + h)

Given the function f(x) = 3/x, we need to find f(x + h) by substituting x + h for x in the function:

f(x + h) = 3 / (x + h)

This step is straightforward and involves a simple substitution. It lays the groundwork for the subsequent steps in the process. The importance of accurate substitution cannot be overstated, as any error in this step will propagate through the rest of the calculation.

Step 2: Calculate f(x + h) - f(x)

Next, we subtract f(x) from f(x + h):

f(x + h) - f(x) = [3 / (x + h)] - [3 / x]

To simplify this expression, we need to find a common denominator, which is x(x + h). Rewriting the expression with the common denominator, we get:

f(x + h) - f(x) = [3x - 3(x + h)] / [x(x + h)]

Now, we simplify the numerator:

f(x + h) - f(x) = [3x - 3x - 3h] / [x(x + h)] = -3h / [x(x + h)]

This step involves algebraic manipulation to combine the two fractions into a single expression. The ability to handle fractions effectively is essential for this calculation.

Step 3: Divide by h

Now, we divide the result from Step 2 by h:

[f(x + h) - f(x)] / h = [-3h / [x(x + h)]] / h

This can be rewritten as:

[f(x + h) - f(x)] / h = -3h / [h * x(x + h)]

We can cancel out the h in the numerator and denominator:

[f(x + h) - f(x)] / h = -3 / [x(x + h)]

Dividing by h is a crucial step because it sets up the expression for taking the limit as h approaches zero. Careful simplification is necessary to avoid errors.

Step 4: Evaluate the Limit as h Approaches 0

Finally, we evaluate the limit as h approaches 0:

f'(x) = lim (h->0) [-3 / [x(x + h)]]

As h approaches 0, the term (x + h) approaches x. Therefore, the expression becomes:

f'(x) = -3 / [x(x + 0)] = -3 / x^2

Thus, the derivative of f(x) = 3/x is f'(x) = -3/x^2. This final step involves taking the limit, which is a fundamental concept in calculus. Understanding how limits work is crucial for finding derivatives using first principles.

Conclusion

By applying the definition of the derivative from first principles, we have successfully determined that the derivative of f(x) = 3/x is f'(x) = -3/x^2. This process involves careful algebraic manipulation and a solid understanding of limits. While this method can be more time-consuming than using differentiation rules, it provides a deeper understanding of what a derivative represents. Mastering first principles is essential for anyone studying calculus, as it lays the foundation for more advanced topics and applications. The derivative, f'(x) = -3/x^2, indicates the instantaneous rate of change of the function f(x) = 3/x at any point x. This result has numerous applications, such as finding the slope of the tangent line to the curve at a specific point or analyzing the function's behavior. In summary, using first principles to find derivatives is a powerful technique that deepens one's understanding of calculus and its applications.

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