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The difference quotient is a fundamental concept in calculus, used to describe the rate at which a function changes. To understand and compute the difference quotient, we start with its general formula. This formula is defined as:
\[ \frac{f(x+h) - f(x)}{h} \]
This equation measures the average rate of change of a function, providing insights into how the function behaves between two points: \(x\) and \(x + h\). It's essentially the slope of the secant line connecting these points.
Whenever you see the term 'difference quotient,' think about how a function's output values change relative to its input values over a small interval \(h\). This is crucial for understanding the underlying behavior of functions and is a stepping-stone for derivative calculations.

Function substitution is a core step in calculating the difference quotient. It involves plugging in specific values or expressions into a function. Given an example function like \(f(x) = \frac{2}{3x}\), we need to calculate \(f(x + h)\).
To achieve this, follow these steps:

• Identify the function: \(f(x) = \frac{2}{3x}\).
• Replace \(x\) with \(x + h\) to get \(f(x + h)\).

This means:
\[ f(x + h) = \frac{2}{3(x + h)} \]
Understanding this substitution is vital. It sets the stage for comparing the function's values at two points \(x\) and \(x + h\). Each substitution step brings us closer to simplifying and understanding the overall function behavior.

Simplifying expressions is essential when working with the difference quotient. It allows for a cleaner and more manageable form. Once we substitute \(f(x)\) and \(f(x + h)\) into the difference quotient, we end up with:
\[ \frac{\frac{2}{3(x+h)} - \frac{2}{3x}}{h} \]
The next goal is to combine and simplify the fractions within the numerator. Start by identifying a common denominator. For this example, the common denominator for \(3(x+h)\) and \(3x\) is \(3x(x+h)\).
We then rewrite the numerator:
\[ \frac{ \frac{2 \times 3x - 2 \times 3(x+h)}{3x \times 3(x+h)} }{h} = \frac{ 2(3x - 3(x+h))}{3x(x+h)} \]
By simplifying the terms inside the parentheses, you get:
\[ 2(3x - 3x - 3h) = -6h \]
This results in the expression:
\[ \frac{-6h}{3x(x+h)} \]
Simplified expressions help us see the core structure of the equation, making it easier to understand and proceed with further calculations.

Finding a common denominator is a critical technique when dealing with fractions, particularly in the difference quotient. The goal is to have a single denominator so that the fractions can be easily combined. For the difference quotient:
\[ \frac{\frac{2}{3(x+h)} - \frac{2}{3x}}{h} \]
To combine these fractions, identify their common denominator. Here, the denominators are \(3(x+h)\) and \(3x\), resulting in the common denominator being \(3x(x+h)\).
This lets us rewrite the expressions within the numerator:
\[ \frac{ 2(3x - 3(x+h))}{3x(x+h)} \]
Now that both terms share a common denominator, we can easily simplify the expression. The numerator simplifies further to:
\[ \frac{-6h}{3x(x+h)} \]
Understanding how to find and use common denominators is crucial for algebra and calculus operations. It streamlines the calculations, ensuring clarity and ease in simplifying complex fractions.