Problem 24 - Calculus

Question

Find a formula for the derivative of the function using the difference quotient. $$g(x)=4 x-5$$

Step-by-Step Solution

Answer

The derivative of \(g(x) = 4x - 5\) is 4.

Step 1: Write the Difference Quotient

The difference quotient is a standard method for finding the derivative of a function. It is given by the expression: equation\frac{g(x+h)-g(x)}{h}.equation This represents the average rate of change of the function over a small interval \((x, x+h)\).

Step 2: Substitute the Function into the Difference Quotient

Using the function \(g(x) = 4x - 5\), substitute \(g(x)\) and \(g(x+h)\) into the difference quotient: equation\frac{(4(x+h) - 5) - (4x - 5)}{h}.equation

Step 3: Simplify the Expression

Simplify the expression within the numerator: \(4(x + h) - 5\) becomes \(4x + 4h - 5\). Substituting this back into the difference quotient, you get: \(\frac{(4x + 4h - 5) - (4x - 5)}{h}\).

Step 4: Combine Like Terms

Combine like terms in the numerator: The \(4x\) terms cancel each other out, and the \(-5\) terms also cancel. You are left with \(\frac{4h}{h}\).

Step 5: Simplify the Result

Divide the numerator and denominator by \(h\): \(\frac{4h}{h} = 4\). This is the derivative of the function \(g(x) = 4x - 5\).

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus used to find derivatives. It serves as the bridge between algebra and calculus, encapsulating how functions change. The standard difference quotient form is \(\frac{f(x+h) - f(x)}{h}\). This expression calculates the average rate of change of a function as you move a small distance \(h\) along the x-axis.

This is especially important for understanding how a function behaves locally around a given point. By analyzing how small changes in \(x\) alter \(f(x)\), the difference quotient provides insights into the function's gradient or slope.

Linear Function

A linear function is a type of function with a constant rate of change, represented by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. These functions produce straight lines when graphed. In this context, the function \(g(x) = 4x - 5\) is linear, showcasing a consistent slope of 4.

Linear functions are essential in mathematics due to their simplicity and clear representation of proportional relationships. They arise frequently in various fields, from physics to economics, often describing basic real-world phenomena like constant speed or rates.

Simplification

Simplification in mathematics is the process of making an expression easier to understand and solve. It generally involves combining like terms, reducing fractions, and canceling out components when possible. During differentiation, simplification is vital to obtaining a clear and manageable expression of the derivative.

For instance, in finding the derivative of \(g(x) = 4x - 5\) using the difference quotient, simplification reduced the complex fraction to 4. This is essential because it provides the exact slope of the linear function efficiently, ensuring clarity and ease.

Rate of Change

The rate of change in a function describes how one quantity changes in relation to another. It can be understood as the slope of the graph of a function. In calculus, the derivative is the primary tool for measuring this rate.

For a linear function like \(g(x) = 4x - 5\), the rate of change is constant. Here, it is 4, meaning for every increase of 1 in \(x\), \(g(x)\) increases by 4. Understanding this concept helps predict behavior and solve practical problems involving changes, such as speed, growth rates, and more.