Understanding the First Principle of Derivatives: A Step-by-Step Guide

Derivatives are a fundamental concept in calculus that help us understand the rate of change of functions. One of the most basic methods to calculate a derivative is by using the First Principle of Differentiation. This article will walk you through the steps to find the derivative of the function y 3x^2 - 2x - 3 and another function y 2x^2 - 3x - 2 using the First Principle. We will also discuss how to use this method accurately and effectively.

What is the First Principle of Differentiation?

The First Principle of Differentiation, also known as the limit definition of the derivative, allows us to find the derivative of a function at any given point. The formula is given by:

[ f'(x) lim_{h to 0} frac{f(x h) - f(x)}{h} ]

This formula represents the average rate of change of the function between two points, which approaches the instantaneous rate of change (derivative) as the distance between the points (h) approaches zero.

Step-by-Step Guide to Using the First Principle

Let's break down the steps to find the derivative of the function y 3x^2 - 2x - 3 using the First Principle.

Step 1: Define the Function

Our function is:

[ f(x) 3x^2 - 2x - 3 ]

Step 2: Calculate f(x h)

Next, we need to calculate f(x h):

[ f(x h) 3(x h)^2 - 2(x h) - 3 ]

Expanding this:

[ 3(x^2 2xh h^2) - 2x - 2h - 3 ] [ 3x^2 6xh 3h^2 - 2x - 2h - 3 ]

Step 3: Find f(x h) - f(x)

Now we compute f(x h) - f(x):

[ f(x h) - f(x) (3x^2 6xh 3h^2 - 2x - 2h - 3) - (3x^2 - 2x - 3) ]

Simplifying this, we get:

[ 6xh 3h^2 - 2h ]

Step 4: Divide by h

Next, we divide by h:

[ frac{f(x h) - f(x)}{h} frac{6xh 3h^2 - 2h}{h} ]

Assuming h eq 0, we can simplify:

[ 6x 3h - 2 ]

Step 5: Take the Limit as h → 0

Finally, we take the limit as h approaches 0:

[ f'(x) lim_{h to 0} (6x 3h - 2) 6x - 2 ]

Therefore, the derivative of the function y 3x^2 - 2x - 3 using the First Principle is:

[ frac{dy}{dx} 6x - 2 ]

Alternative Examples and Practical Applications

Let's apply the First Principle to another function: y 2x^2 - 3x - 2.

Using the First Principle on y 2x^2 - 3x - 2

Our function is:

[ f(x) 2x^2 - 3x - 2 ]

First, we calculate f(x h):

[ f(x h) 2(x h)^2 - 3(x h) - 2 ]

Expanding this:

[ 2(x^2 2xh h^2) - 3x - 3h - 2 ] [ 2x^2 4xh 2h^2 - 3x - 3h - 2 ]

Now we compute f(x h) - f(x):

[ f(x h) - f(x) (2x^2 4xh 2h^2 - 3x - 3h - 2) - (2x^2 - 3x - 2) ]

Simplifying this, we get:

[ 4xh 2h^2 - 3h ]

Dividing by h:

[ frac{f(x h) - f(x)}{h} frac{4xh 2h^2 - 3h}{h} ]

Assuming h eq 0, we can simplify:

[ 4x 2h - 3 ]

Finally, we take the limit as h → 0:

[ f'(x) lim_{h to 0} (4x 2h - 3) 4x - 3 ]

Therefore, the derivative of the function y 2x^2 - 3x - 2 using the First Principle is:

[ frac{dy}{dx} 4x - 3 ]

Conclusion: The First Principle of Differentiation provides a comprehensive way to understand and calculate derivatives, offering a deeper insight into the rate of change of functions.